Algebra: Modeling Our World...6 In addition, this course is designed to provide more relevant...

Algebra: Modeling Our World

Michael CatalanoDakota Wesleyan University

Spring, 2009 edition

Contents

1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52: Representing Quantitative Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Reading Questions for Representing Quantitative Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26Representing Quantitative Information: Activities and Class Exercises . . . . . . . . . . . . . . . . . . . . 31

3: Collecting and Analyzing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40Reading Questions for Collecting and Analyzing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58Collecting and Analyzing Data: Activities and Class Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4: Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Reading Questions for Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Functions: Activities and Class Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5: Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Reading Questions for Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Linear Functions: Activities and Class Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6: Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Reading Questions for Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141Exponential Functions: Activities and Class Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

7: Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151Reading Questions for Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Logarithmic Functions: Activities and Class Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

8: Power and Quadratic Functions/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169Reading Questions for Power and Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .183Power and Quadratic Functions: Activities and Class Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 187

9: Transformations of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192Reading Questions for Transformations of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199Transformations of Functions: Activities and Class Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

10: Other Families of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207Reading Questions for Other Families of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218Other Families of Functions: Activities and Class Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

11: Adding Things Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226Reading Questions for Adding Things Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .237Adding Things Up: Activities and Class Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

12: Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243Reading Questions for Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257Probability: Activities and Class Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .261

Appendix 1: Miscellaneous Mathematics Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

Appendix 2: Data Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

Appendix 3: How Large is Large? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

Algebra: Modeling Our World

1. Introduction

Example is the school of mankind. He will learn at no other.— Edmund Burke

Example is not the main thing in influencing others. It is the only thing.— Albert Schweitzer (1875 - 1965)

You have very likely taken an algebra course or two in your time, and this text is designed foruse in a college algebra course or what might be termed quantitative literacy courses at a similarlevel. You will likely find a good deal which will be familiar.

However, courses using this text are also likely to be different than other algebra courses youmay have taken. In particular,

1. This course will emphasize learning through example. Information will often be presentedin the context of real-world situations. Many of our examples will be related to social andeconomic issues, including, hunger, poverty, energy, and the environment both within theUnited States and worldwide.

2. A large part of the work you do in this course will be done in groups, including much of thedaily homework.

3. This course is designed to help you become a better learner, especially of mathematics. Inour society, knowledge is expanding at an ever increasing rate, and the knowledge which isrelevant and useful today may not be tomorrow. This means that a college education is nolonger only about acquiring “all” the knowledge you need to know to be an educated citizen.Rather, it is about acquiring a base of useful knowledge and examples, and the ability tobuild on this base as you progress through life. We will explicitly address this issue as partof the course.

4. The text is in more of a narrative style than most other mathematics textbooks. Althoughthere will be examples and problems worked out in the text, the book is designed to be read‘cover to cover’ rather than used only as a reference. I encourage to read thoroughly and readahead. There are approximately 160 pages of text to read, not counting exercises and groupactivities. In a 15 week semester, you will be roughly reading 10 to 12 pages each week.

5. This course will emphasize the usefulness of mathematics. While mathematics is a worthy(and beautiful!) discipline in its own right, we will concentrate on what mathematics can sayabout the real-world, and help you develop skills in applying mathematics to understandingreal-world situations.

The overall goal is for you to not only develop your mathematical knowledge and skills, but tobecome more confident in those skills, gain a greater appreciation for the usefulness of these skillsin understanding aspects of our modern society, and enhance your ability to apply those skills inwhatever your chosen field happens to be.

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In addition, this course is designed to provide more relevant content and a different pedagogicalexperience for future elementary school teachers.

As technology has “mathematicized” the workplace and as mathematics has permeatedsociety, a complacent America has tolerated underachievement as the norm for math-ematics education. We have inherited a mathematics curriculum, conforming to thepast, blind to the future, and bound by a tradition of minimum expectations—- from Everybody Counts: A report to the nation on the future of mathematics edu-cation (Washington, DC: National Research Council/Mathematical Sciences EducationBoard, 1989)1 (from Summary)

Mathematics in the Real World

Providing nourishment, health, and happiness to all of God’s children will cost eachAmerican only a few dollars. That’s a small price to pay. In the long run it will cost usa lot more if we let 300 million children remain hungry and poorly educated. How manyAlbert Einsteins, Marie Curies, Martin Luther Kings, and Michael Jordans will neverdevelop if we do nothing to feed the hungry children of the world? Who can evaluatethat loss?— Ambassador George P. McGovern

As noted earlier, this course will emphasize examples from the real world, including examplesrelated to issues of hunger and poverty. Why this particular focus? There are a number of reasons.

The first is that hunger and poverty are prevalent throughout the world and directly affectbillions of people. Even in the United States, one of the wealthiest nations on earth, a significantproportion of the population has to cope with not having sufficient resources for basic necessitieson a monthly, if not daily, basis. The author would not be surprised if there are people readingthis introduction who feel they have experienced this. The second is that these issues are relatedto many other aspects of our world wide society, including agriculture, energy, political democracy,the environment, education, and even violent conflicts including terrorism. While we may notthink about these issues on a day to day basis, they greatly influence the larger context of ourlives, from what types of jobs and other opportunities might be available to us, to where we liveand the quality of life that we experience there.

Finally, this focus partially arose out of the mission of the McGovern Center for Leadership andPublic Service at Dakota Wesleyan University.2 The center is named after former South DakotaSenator George McGovern, a Dakota Wesleyan University graduate, and probably most famous forhis 1972 campaign for President of the United States against Richard Nixon. McGovern has spentmuch of his career working to alleviate world hunger, including recent service as United NationsGlobal Ambassador on World Hunger.

Although hunger and poverty issues will be emphasized throughout the course, this will certainlynot be the sole focus. We will consider quite a variety of examples and applications of mathematics.Here are a few of the issues and situations we will be considering.

1. Baby boom: The so-called “baby boom” is the result of 75 million babies being born in the18 years from 1946 to 1964, as compared to 44 million from 1926 to 1946.3 This populationsurge has had, and will continue to have profound effects on American society at least through2050. For example, in 1950, there were 3 persons 85 years and up for every 100 persons age50 to 64. In 2050, this ratio is projected to increase to 27 to 100. In the year 2030, there willbe as many 70-year-olds as 30-year-olds (up from 1 to 3 in 1990). The number of workingage citizens for every person over 65 is projected to go from 3.2 to 2 by 2030.

1http://www.napanet.net/ jlege/allcount.htm2http://www.mcgoverncenter.com/3http://www.hometechsystems.com/demographics2.htm

1. Introduction 7

How are such projections made, and how confident can we be of these projections? Assumingthese projections prove true, what sort of impact will this have on our society? Are thereother countries where “baby booms” have occurred or are occurring? How differently willyou have to plan for retirement than your parents or grandparents?

2. Ozone Ouch: The ozone hole over the Antarctic grew from about 180,000 square miles in1980 to about 9 million square miles in 2001. How can we use the known data to predictwhat might happen in the future? What assumptions are we making in our projections? Howreasonable are these assumptions?

3. It’s Never Too Early to Think About Retirement: What would you like to be doingwhen you are 65 or 70? Whether you have ambitious plans (a series of world cruises?) or not,you will need to set aside a certain amount of resources to live on, unless you are counting onworking until the day you die. There is the Social Security System, but as noted in item oneabove, there will be fewer and fewer workers to support retirees as time goes on, so SocialSecurity benefits may be diminished by the time you are of retirement age. How much shouldyou be saving for your retirement? When should you start?

4. Demographic Details: As part of our consideration of hunger and poverty issues, we willbe looking at a variety of world demographic data. For example, the chart below shows theannual population growth rate worldwide since 1950, and projected out until 2050.How are these future predictions made? What assumptions are part of making these projec-tions? What will be the effect on the total population? What happened around 1960?

Figure 1: World population growth rates

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5. Medical Testing: You have tested positive for a dangerous, though not too common disease.How concerned should you be? (In other words, what are the chances you actually have thedisease?)

6. The Great California Recall: In October of 2003, action hero Arnold Schwarzenegger waselected Governor of California, replacing Governor Grey Davis who was removed using therecall provision of California state law. In one poll of 801 registered California voters leadingup to the election, 50% indicated they would vote to remove Governor Gray Davis fromoffice, and Democratic Lieutenant Governor Cruz Bustamante led the pack of replacementcandidates with 35%, with Schwarzenegger favored by 22% of the voters. The margin of errorfor the poll was reported as 3%.How do we find such “margins of error,” and what do they really tell us? More generally,how can we use sampling appropriately to make predictions? How did Schwarzenegger endup getting elected?

Hunger and Poverty

We all have some awareness of the problem of world hunger, and you may have some ideasabout why it occurs, and what can or cannot be done about it. This course will provide you withthe opportunity to study some aspects of hunger and poverty in more detail, and to see how muchof the “conventional wisdom” is really true. Here are some questions we might consider as we thinkabout world hunger. These are difficult questions, and even with the help of mathematics, it mayonly be possible to provide incomplete answers.

Is hunger really just a matter of there not being enough food in particular locations atparticular times, or is the problem more complicated?

Does the world currently produce enough food to feed everyone? If so, how many morepeople can world resources reasonably support? How long before the world populationreaches this level?

Could we eradicate hunger by simply giving enough food to those who are hungry? Ifwe tried this, how much food would donor nations have to provide on an annual basis?

Is hunger simply a consequence of some being rich and others being poor? Could wealleviate hunger by making the world society more equitable, from an income or wealthstandpoint?

Is “free trade” a good way to alleviate hunger? Would hunger disappear if we were ableto establish a completely free global market system?

Can we alleviate hunger and still maintain reasonable environmental standards world-wide, or will we have to sacrifice environmental concerns in order to feed everyone?

As we consider questions like these, we will see how quantitative information and quantitativemethods, in other words mathematics, can help us better understand what is happening, and whatmay happen in the future.

George McGovern

The Biographical Directory of the United States Congress provides the following informationon George McGovern.4 McGovern served in the U.S. Senate from 1963 to 1981, and the U.S. Houseof Representatives from 1957 to 1961.

4http://bioguide.congress.gov/scripts/biodisplay.pl?index=M000452

1. Introduction 9

McGOVERN, George Stanley, a Representative and a Senator from South Dakota; bornin Avon, Bon Homme County, S.Dak., July 19, 1922; attended the public schools ofMitchell, S.Dak., and Dakota Wesleyan University, 1940-1942; enlisted in the UnitedStates Army Air Corps in June 1942, flew combat missions in the European Theater,and was discharged from the service in July 1945; returned to Dakota Wesleyan Univer-sity and graduated in 1946; held teaching assistantship and fellowship at NorthwesternUniversity, Evanston, Ill., 1948-1950, receiving his Ph.D. from that university in 1953;professor of history and government at Dakota Wesleyan University 1950-1953; execu-tive secretary of South Dakota Democratic Party 1953-1956; member of Advisory Com-mittee on Political Organization of Democratic National Committee 1954-1956; electedas a Democrat to the Eighty-fifth and Eighty-sixth Congresses (January 3, 1957-January3, 1961); was not a candidate for renomination in 1960, but was unsuccessful for electionto the United States Senate; appointed special assistant to the President January 20,1961, as director of the Food for Peace Program, and served until his resignation July18, 1962, to become a candidate for the United States Senate; elected to the UnitedStates Senate in 1962; reelected in 1968 and again in 1974, and served from January 3,1963, to January 3, 1981; chairman, Select Committee on Unmet Basic Needs (Nineti-eth Congress), Select Committee on Nutrition and Human Needs (Ninety-first throughNinety-fifth Congresses); unsuccessful candidate for reelection to the U.S. Senate in1980; unsuccessful candidate for Democratic presidential nomination in 1968 and 1984;unsuccessful Democratic nominee for President of the United States in 1972; lecturerand teacher; U.S. Ambassador to the United Nations Food and Agricultural Agencies inRome, Italy, 1998-2001; United Nations Global Ambassador on World Hunger, 2001-.

Acknowledgements

This book owes a great deal to Understanding Our Quantitative World, published by the Mathe-matical Association of America, and its authors Janet Andersen and Todd Swanson of Hope Collegein Holland, Michigan. The incorporation of social issues, collaborative learning, and an emphasison reading in this text are all informed by their work. I am very grateful to Janet and Todd fortheir help and encouragement in developing this text.

I also would especially like to acknowledge Janet for our many years of friendship, the manyconversations we had during the 2003-2004 academic year while I was on sabbatical at Hope College,and for her help in arranging for my sabbatical appointment at Hope.

Very tragically, Janet was killed on Thanksgiving Day of 2005 in an automobile accident. Iam dedicating this work to her. She has been greatly missed by many people, including myself,and leaves not only the mathematical community, but also our larger human family, poorer in herabsence. She was a truly outstanding person.

I am grateful for the extensive support I have received from the National Science Foundation.Firstly, NSF support through the AIRE program allowed work to begin on Algebra: ModelingOur World during my 2003-2004 sabbatical year. Secondly, the NSF has supported continuingwork on this project through a grant under the Course Curriculum and Laboratory Improvement- Educational Materials Development program (CCLI-EMD award number 0442979) to DakotaWesleyan University, for which the author is the Principal Investigator. I would like to acknowledgethe help that Todd Swanson, former students Nikki Hobbie and Kristi Kogel, and also my colleaguesDr. David Mitchell and Mr. Kevin Lein have provided this project. In addition, I would like toacknowledge the suggestions and input of many students who participated in pilot sections usingthis text at Dakota Wesleyan.

Finally, I would like to acknowledge Dr. Rocky VonEye and Dr. Mike Farney for their incrediblesupport and wonderful friendship during my years at Dakota Wesleyan. I don’t think I could havebetter mathematical colleagues.

2. Representing Quantitative InformationNumeracy is a proficiency which is developed mainly in mathematics but also in othersubjects. It is more than an ability to do basic arithmetic. It involves developing con-fidence and competence with numbers and measures. It requires understanding of thenumber system, a repertoire of mathematical techniques, and an inclination and abilityto solve quantitative or spatial problems in a range of contexts. Numeracy also demandsunderstanding of the ways in which data are gathered by counting and measuring, andpresented in graphs, diagrams, charts and tables.— from National Numeracy Strategy published by the United Kingdom’sDepartment for Education and Skills5

We see quantitative information in a variety of forms and contexts nearly every day. In this chap-ter, we will consider two ways in which quantitative information is commonly presented, namelyas graphs and as tables. Each of these has its own advantages and disadvantages. Employing morethan one form at a time often greatly enhances our understanding of a particular situation andour ability to solve problems. In Chapter Four, we will consider two additional ways to representquantitative information, in symbols or equations, and in verbal statements.

As will be the case throughout the book, we will often introduce concepts via real-world exam-ples.

Energy Supplies

Our ignorance is not so vast as our failure to use what we know.— M. King Hubbert6

Energy issues are almost constantly in the news, and are now even becoming the subject ofHollywood films (e.g. Syriana, released in 2005 and starring George Clooney). The price at thepump for gasoline (nearing a national average of $4 per gallon in May of 2008), the cost of heatingduring a hard winter, the diminishing supply of fossil fuels, and battles between environmentalistsand the energy industry over drilling in environmentally sensitive areas are all recurrent themes inpolitical discussions.

Energy supply and cost is increasingly an issue with respect to hunger. During the 20th century,agriculture evolved from an activity predominantly dependent upon nature and human labor into alargely technological enterprise dependent on academic research, chemical inputs, and a high use offossil fuels. As developing countries modernize their agricultural practices, they become dependenton fossil fuels and put greater demands on the world energy supply. As known reserves of oil andother fossil fuels are used up, prices are likely to increase, making it harder for farmers, especiallysmaller farmers, to remain economically viable. How much longer can we expect the earth’s fossilfuel supplies to last?

In 1956, Dr. M. King Hubbert predicted that the volume of oil produced annually in the U.S.would increase more slowly over the coming 15 years, and peak around 1970. After this point,annual oil production would begin decrease. At the time, his prediction was greeted with deepskepticism, even ridicule. Figure 1 shows a graph of the actual data, from 1954 to 2002.7

In the graph, we can visually estimate the total production for any given year. For example, in1970 the production was approximately 9,900 thousand barrels, or equivalently 9.9 million barrels,per day. We would say that the value of U.S. Oil Production in 1970 was about 9,900 thousandbarrels per day.

From the graph, we can not only see that Hubbert’s prediction was dead on, but also get anidea of how large that maximal production was, and how much production has dropped off sincethen.

5http://www.standards.dfes.gov.uk/primary/publications/inclusion/1152065/6This quote and other information on Hubbert’s work is from www.hubbertpeak.com7The data on which this graph is based is from the Energy Information Administration of the Department of

Energy at http://www.eia.doe.gov/emeu/aer/txt/ptb0502.html.

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2. Representing Quantitative Information 11Total U.S. Oil ProductionThousands of Barrels per Day02,0004,0006,0008,00010,00012,0001950 1960 1970 1980 1990 2000 2010Figure 1: Total U.S. Production of Crude Oil

Hubbert made his forecast assuming that oil is a finite resource, and that production startsat zero and increases over time until a peak production level is reached. Once the peak has beenpassed, production declines until the resource is depleted. Since the 1950’s, so-called Hubbertcurves have been used to study production in every oil-producing country, the world at large, aswell as for other finite resources (e.g. coal). Here is another production graph, which predicts thatworld oil production will peak around 2010.8

Figure 2: World Oil and Gas Production

8Graph from the The Association for the Study of Peak Oil and Gas (ASP), http://www.asponews.org/.

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One of the implications of Hubbert’s work is that, once we reach the peak, we will have usedapproximately half of the earth’s total oil reserves. If the peak occurs in 2010, it will have takenabout 150 years to go from zero production to depleting half of the resource.9 By comparingproduction estimates with current trends in energy consumption, we will be able to estimate howlong the remaining half of the reserves might last. Even without any data, we can reasonablyassume that we are using energy at a much greater rate today than in years past, and so there issome reason to believe reserves will not last even close to 150 years.

These graphs provide us with an easily understandable picture of U.S. and world oil production.One could certainly use these to make the case that a global energy crisis looms in the not toodistant future. The importance and usefulness of graphs like these are that they greatly assist usin understanding quantitative information, and reaching conclusions based on that information.

A whole essay might be written on the danger of thinking without images.— Samuel Taylor Coleridge

What Makes a Good Graph?

Not all graphs are created equal. Good graphs are constructed with several considerations inmind.

A graph should indicate the variables which are being displayed. In Figure 1, Total U.S. OilProduction is indicated as the dependent variable, and one can infer from the graph that theindependent variable is time measured in years. Unless otherwise noted, the dependent variablevalues are indicated along the vertical axis and the independent variable is indicated along thehorizontal axis. This is standard for two-variable graphs. We could have also titled the graph“U.S. Oil Production Versus Time” or “U.S. Oil Production over Time” to indicate time as theindependent variable and production as the dependent variable. Sometimes it is useful to labeleach of the axes with the appropriate variable. In Figure 2, the vertical axis is labelled Gb/a,which is short for “gigabarrels annually” or “billions of barrels per year.”

The scale on both the vertical and horizontal axes is given and is uniform. In other words,marks are made at regular, evenly-spaced intervals along each axis and each mark represents thesame number of units. Note that the scale and units on one axis could certainly be different thanthat on the other axis. In Figure 2, the scale on the horizontal axis is ten years, and the scale onthe vertical axis is 10 Gb/a.

Choosing an appropriate window for the graph is important. The window is given by theminimum and maximum values for both the horizontal and vertical scale. In Figure 1, we wouldsay the window is 1950 ≤ Year ≤ 2010, 0 ≤ Oil Production ≤ 12, 000. The horizontal baselineof the graph is 1950, and the vertical baseline is 0. The main goal in choosing a window is toallow the general characteristics of the data (or equation) depicted to be easily seen. All the datavalues should be included without including a huge amount of empty space. In Figure 1, we couldhave used a vertical baseline of 4,000 instead of 0, as this still would have easily encompassed allof our data.

Types of graphs

The graphs in Figures 1 and 2 are called a line graphs. Such graphs consist of a (usuallyunbroken) “line” or curve. Although “line” usually implies straightness, in this context it does not.Line graphs can also include designated points which are connected by the line, as in Figure 1. Agraph can certainly include several line graphs at the same time, as in Figure 2, where we haveseparate line graphs for oil, gas, coal, total production, etc.

Figure 3 shows an example of a bar graph. The independent variable is time, and the depen-dent variable is the percentage of U.S. citizens who report ever having used marijuana. Bar graphsare often used when there are not too many values for the dependent variable.

9According to data provided by Jean Laherrre at the ASPO website, the first oil wells in the U.S were drilledaround 1860.

2. Representing Quantitative Information 13

For example, if we had used a bar graph instead of a line graph to represent U.S. Oil Productiondepicted in Figure 1, we would have had to choose between having nearly 50 bars, one for eachyear, or skipping some years so as to reduce the number of bars.

0%

5%

10%

15%

20%

25%

30%

35%

40%

9 12 15 18 21 22 23 24 25 26 29 30 31

Percent of US. Citizens Who Have Ever Used Marijuana Vs Year from 1970

Figure 3: A bar graph of Marijuana Use in the U.S.

U.S. Average Gasoline Prices - Regular

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

1970 1975 1980 1985 1990 1995 2000 2005 2010

Year

Pri

ce in

Do

llars

Figure 4: Graph of U.S. gasoline prices

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Finally, Figure 4 depicts what is called a scatterplot. As in Figure 1, the independent variableis years. Here, instead of “connecting the dots” with a line, we simply plot one point for each year.The lack of a line does not keep us from seeing any general trends in what is happening. Note thatthe window is 1970 ≤ Year ≤ 2010, 0 ≤ Gas Prices ≤ 3.50.

Representing Quantitative Information Numerically

When a set of data is presented in the form of a table, we refer to this as a numericalrepresentation. For example, Table 1 provides the same information on gasoline prices as in Figure4, but in numerical form.10

Table 1: Unleaded Gasoline Prices in the U.S.

Year 1976 1977 1978 1979 1980 1981 1982 1983 1984Price 0.61 0.66 0.67 0.90 1.25 1.38 1.30 1.24 1.21Year 1985 1986 1987 1988 1989 1990 1991 1992 1993Price 1.20 0.93 0.95 0.95 1.02 1.16 1.14 1.13 1.11Year 1994 1995 1996 1997 1998 1999 2000 2001 2002Price 1.11 1.15 1.23 1.23 1.06 1.17 1.51 1.46 1.36Year 2003 2004 2005 2006 2007 2008Price 1.59 1.88 2.30 2.59 2.81 3.19

While the associated table gives the precise information, the graph gives you a better pictureof how prices have changed over time. A main advantage of describing data in graphical form isthat it allows you to observe trends and see overall behavior quickly and easily, much more sothan tables or equations. On the other hand, tables often take up less space. Finally, in manycases quantitative information starts as data in numerical form. In this form, it can be efficientlystored or transmitted electronically, transported between different software packages, and easilymanipulated or statistically summarized, as we will see in the next chapter. For example, the gasprices data can be stored numerically using 4 kilobytes (kb) of computer space, while the graphshown takes up over 28 kb.

Interpreting Quantitative Information: An Example

Displaying data in tables or graphs is not an end in itself. Rather, we display data in order tohelp us answer questions, reach conclusions, or solve problems. Often, we collect and display datain order to answer a particular questions. Here is one example.

To what extent do countries with a large percentage of poor people also have a largepercentage of hungry people?

We might reasonably guess that the answer to this question is “to a (very) large extent.” Thisquestion concerns the relationship between two variables. Table 2 shows these two variables, alongwith the country labels for the data. The first variable, denoted Poverty Rate in the table, is moreprecisely the National Poverty Rate given as the percent of the total population defined as livingin poverty in that country.11 The second variable is the percentage of children under five who aredefined as being severely underweight for their age. This percentage is one of several common waysto measure the extent of malnutrition in a country, or among children in a country. 12

10Data from the Energy Information Administration at http://www.eia.doe.gov. Data for 2008 is the average ofmonthly averages through April.

11Poverty Data is available from the World Bank.12Malnutrition Data is from UNICEF, http://www.childinfo.org/eddb/malnutrition/index.htm


Table 2: National Poverty Rates and Percentage of Severely Underweight Children Under Five.

Pov. Rate Underwt. Pov. Rate Underwt.Albania 25.4 4.3 Kazakhstan 34.6 0.4Algeria 12.2 1.2 Kenya 42 6.5Armenia 53.7 0.1 Kyrgyz Republic 64.1 1.7

Azerbaijan 49.6 4.3 Lao PDR 38.6 12.9Bangladesh 49.8 13.1 Lesotho 49.2 4

Benin 33 7.4 Madagascar 71.3 11.1Bolivia 62.7 1.7 Malawi 65.3 5.9

Bosnia and Herzegovina 19.5 0.6 Malaysia 15.5 1.2Brazil 17.4 0.6 Mauritania 46.3 9.2

Burkina Faso 45.3 11.8 Mauritius 10.6 2.2Burundi 36.2 13.3 Mexico 10.1 1.2

Cambodia 36.1 13.4 Mongolia 36.3 2.8Cameroon 40.2 4.2 Morocco 19 1.8

Chad 64 9.8 Mozambique 69.4 9.1Colombia 64 0.8 Nepal 42 12Costa Rica 22 0.4 Nicaragua 47.9 1.9

Cote d’Ivoire 36.8 4 Niger 63 14.3Djibouti 45.1 5.9 Nigeria 34.1 10.7

Dominican Republic 28.6 1 Pakistan 32.6 12.8Ecuador 35 1.9 Peru 49 1.1Egypt 16.7 2.8 Romania 21.5 0.6

El Salvador 48.3 0.8 Rwanda 51.2 7.1Eritrea 53 17 Senegal 33.4 4.1Ethiopia 44.2 16 Sierra Leone 68 8.7Georgia 11.1 0.2 Sri Lanka 25 4.8Ghana 39.5 5.2 Tanzania 35.7 6.5

Guatemala 56.2 4.7 Togo 32.3 6.7Guinea 40 5.1 Tunisia 7.6 0.6

Guinea-Bissau 48.7 5.2 Uganda 55 6.7Honduras 53 4 Ukraine 31.7 0.5Hungary 17.3 0.2 Uzbekistan 27.5 5

India 28.6 18 Venezuela, RB 31.3 0.7Indonesia 27.1 8.1 Vietnam 50.9 5.8Jordan 11.7 0.5 Yemen, Rep. 41.8 14.5

Zimbabwe 34.9 1.5

16

Perhaps the first thing one notices about this table is that this is a lot of data! In fact, thereare 69 countries listed, with the data given in alphabetical order by country. Both the amount ofdata and the way it is organized in the table make it a little hard to make any general conclusions.We could certainly pick out of few countries and see if those with a low(high) Poverty Rate alsohave a low(high) Underweight Rate. However, this is a little awkward and we could also easilyreach the wrong conclusion depending on which countries we happened to pick. What we need isa quick and easy way to see if our assumption about the effect of poverty on malnutrition reallyseems to be true. Here we can use a scatterplot to good advantage.

Recall that standard practice is to put the independent variable on the horizontal axis and thedependent variable on the vertical axis. In this example, we would probably think of malnutritionas depending on the poverty rate. Given this, we would have Poverty Rate as our independentvariable and put it on the horizontal axis and Underweight Rate as the dependent variable andput it on the vertical axis.13

A scatterplot of the data from Table 2 is shown in Figure 5. From this scatterplot, you cansee that, although the points are widely scattered, it does seem the points are generally gettinghigher the further to the right you go. This is an example of a positive association. Two variableshave a positive association when an increase in the value of the independent variable leads toan increase in the value of the dependent variable. In our situation, this means that countries withhigher poverty rates tend to have more malnutrition. Since there is a lot of scatter, the positiveassociation is weak.

Percent Severely Underweight Children Vs National Poverty Rate

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Figure 5: A scatterplot of Percent of Severely Underweight Children Versus Poverty Rate.

What more could we say about our poverty and malnutrition data from the scatter plot inFigure 5? Spend a couple minutes considering the plot. You may want to look at particularpoints or groups of points and see if you can find the corresponding countries in the table. Whatobservations do you make?

(You may want to refer back to this example when doing the activities for this chapter)

Here are a few examples of observations that one might make. This is not meant to be a completelist, or even a list of “what you should have observed.” The observations are simply meant to give

13It is not always clear in all cases which of two variable should be the dependent and which the independent.Often it is a judgement call, but at other times there may be a fairly obvious choice.


you examples of the types of things you might look for when considering a scatterplot.One observation is that the data almost seems to fit into sort of a triangular or pie-piece pattern,

except for a single point (the highest point in the plot). There are points clustered close to thehorizontal axis all through the plot, and there is also a very linear increasing pattern along the topof the plot. Perhaps we could say the following.

A low poverty rate implies a low malnutrition rate, but a high poverty rate does notnecessarily mean a high malnutrition rate, as many countries with higher poverty ratesalso have low malnutrition rates.

Going back to the high point of the graph, we might wonder what is special about this country.The point seems “far away” in some sense from the rest of the points, and we might classify itas an outlier. With a little searching, you could see from the table that this country is India.Why would India have such a high malnutrition rate, especially as its poverty rate is not thathigh? You might know that India is the second most populous country on earth (after China).Could the large population be having an effect? We could look for other populous countries in ourtable. Unfortunately, the data we have does not include all countries, and in particular, China,the United States, and the Russian Federation are all populous countries that are not included.Another speculation might be that the data for India is not accurate. We might want to know moreabout how the data was collected. Did UNICEF collect the data, or did some other organizationor the individual countries themselves supply the data? We would be jumping to a conclusion ifwe claimed that the data must be inaccurate just because the given point seems to be an outlier,however, it is sometimes the case that outliers are the result of errors of one sort or another.

One might also notice that the countries with poverty rates below about 25% seem to formtheir own group. All of these countries have low malnutrition rates, and there is sort of a “gap”between these points and the rest of the points. If you ignore these countries (say, by coveringthem up when you look at the scatterplot), and look only at the higher poverty countries, thenthere seems to be almost no association between the two variables. As a hypothesis, we mightconjecture that 25% is a sort of “cutoff point”. Countries that are below this point in poverty havelow malnutrition, and for countries above this point, the poverty rate does not seem to have mucheffect on how much malnutrition there is.

There are certainly other observations one might make. At this point, we are only making someinitial observations and also bringing up further questions for consideration. As we proceed throughthe course, we will discuss how our observations can be made more precise, more “quantifiable”,and how we can better decide which conclusions we can have high confidence in and which aremore speculative.

Negative and Positive Associations

It was previously stated that the graph in Figure 5 had a positive association since the pointsin the graph tended to get higher as we go to the right. In other words, the dependent variabletended to increase as the independent variable increased.

Two variables have a negative association when an increase in the value of the independentvariable leads to a decrease in the value of the dependent variable. An example of this is therelationship between latitude and average January temperature for cities in the United States. Wewould think of temperature as depending on the latitude, so we graph the independent variable,latitude, along the horizontal axis. As the latitude increases (moving farther north), the averageJanuary temperature decreases. Figure 6 graphs the latitude and average January temperature fortwelve cities in the eastern United States.14 Since there is a lot less scatter in this graph than inFigure 5, this graph shows a much stronger association.

To reiterate, one difference between Figure 5 and Figure 6 is that the points in Figure 6 seem toclosely follow a “linear” pattern, while the points in Figure 5 have much more “scatter” to them. Wewould say that the relationship between temperature and latitude is strong while the relationshipbetween poverty and malnutrition is weak, or at least much weaker than the temperature andlatitude relationship. In Chapter Five, we will make this idea of strong and weak more precise. For

14The data was gathered from The World Almanac and Book of Facts 1996, pp 180, 594, 595.

18

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Figure 6: A scatterplot of latitude and average January temperatures for cities in the easternUnited States.

now, the idea is that the more closely the points in a scatter plot cluster around a straight line,the stronger the (linear) relationship between those variables.

Displaying and Interpreting One-variable Data

Scatterplots, like the one for malnutrition versus poverty in Figure 5, help us investigate therelationship between two variables. Often, we wish to consider one variable at a time. As withtwo-variable data, the first step is often deciding on the best way or ways to present or display thedata so that the information contained can be easily understood. In this section, we will look atwhat types of graphs and tables are most appropriate for displaying various types of one-variabledata sets.

The data set listed in Table 3 gives the percentage of females of the appropriate age group whocompleted primary school (elementary school) in 23 countries. Each individual number given iscalled a data point.

This data involve just one numerical variable, the percentage completion rate. The data arelisted from the lowest percentage country to the highest. There is no reason the data have to beordered in this way. Another reasonable choice would be to list the countries alphabetically. Thenames of the countries are examples of what are called (data) labels.

Spend a minute or two looking over the data in Table 3. Make notes of any particularlynoteworthy aspects of the data. What do the data seem to tell you about the state of educationthroughout the world? Are there any questions that arise in your mind about these data?

There are many things we could say about this data set.15 Here are a few examples.

The percentage of girls who complete primary school goes from a low of under 10% to over100%. This is a very wide range, especially given the relatively small number of countries.

One question we might have is “how is it possible to get over 100%?” It would be beneficial

15Data is from the World Bank data base


Table 3: Completion Rates for Primary (Elementary) School, Females, Year 2000

Primary Primary PrimaryCountry Comp. Rate Country Comp. Rate Country Comp. RateChad 8.53 Nepal 58.35 Mexico 93

Guinea-Bissau 23.5 Congo, Rep. 59.68 Guyana 95.15Guinea 24.27 Nigeria 60.66 Korea, Rep. 97.78

Sierra Leone 30 Laos PDR 64.23 Jamaica 98.3Senegal 34.11 Nicaragua 70.04 Uruguay 101.12

Congo, Dem. Rep. 34.21 Bangladesh 71.67 Jordan 105.59Yemen, Rep. 38.28 Trinidad & Tob. 84 Dominica 107Cambodia 51.47 Indonesia 91.71

to have a more complete description of how these percentages were computed. One possible ex-planation is that the number of girls who completed primary school in a given year is more thanthe number of girls who are at the age where they should be completing school because some girlshave taken longer to complete school and are older than the usual completion age.

We might also notice that there are only 23 countries listed. What about all the other countries?Most of the countries are from Africa and Asia. There are no European countries, and very fewindustrialized countries (like the U.S., Great Britain, Japan, Russia, etc.). Were these countriesintentionally left out by the World Bank, or was data simply not available for these countries?Because this group of countries does not seem to have been randomly selected, and is not represen-tative, we cannot use this data to produce (say be averaging) a valid measure of completion ratesfor the world as a whole.

Notice that it looks like more than half of the countries have completion rates over 50%, sothere are fewer countries with very low completion rates.

As with the example discussing poverty and malnutrition, the discussion above is notmeant as a complete “right answer” to what you might notice about this data set.Rather, it is meant to illustrate a way of thinking about data, as well as an attitude ofquestioning. If you did not notice these aspects of the data, or did not ask the samequestions, that is OK for now. As you progress through the course, you should gainskill in becoming more thorough at this kind of analysis and develop a more questioningattitude regarding data. The important thing for now is to begin to reflect on how youcan think about, and learn from, numerical data.

Frequency Distributions

In our examples with two variable data, we found the visualization provided by a scatterplotvery helpful in understanding the data. Similarly, a graph or some other way of summarizing thedata in Table 3 might be helpful.

One common method of organizing data is a frequency distribution, which is a table inwhich the data is grouped into categories (often called classes) and the number of data points ineach of these categories is given. The numbers in Table 3 are percentages. We might choose todivide our data up in categories in 10% intervals, 0%, 10%, 20%, . . . up to say 110%.16 Then, wecount how many data points are in each category. For example, there are 3 countries that would

16There are other ways to group these data. For example, we could go in intervals of 15% or 20%

20

fit between 20% and 30%. Standard practice is for us to account values that are strictly greaterthan 20%, and less than or equal to 30%. Counting the number of percentages in each class andlisting that number as its frequency, we obtain the frequency distribution shown in Table 4.

CompletionRate Frequency

0 to 10 110 to 20 020 to 30 330 to 40 340 to 50 050 to 60 360 to 70 270 to 80 280 to 90 190 to 100 5

100 3

Table 4: A frequency distribution for Primary School Completion Rates.

The frequency distribution gives us a better understanding of how the completion rates aredistributed. For example, we can easily see that seven of the 23 countries have rates below 50%.

Histograms

A histogram is a bar graph of a frequency distribution. In a histogram, one axis (usually thehorizontal) is scaled according to the classes in the frequency distribution. The other axis givesthe frequency in any appropriate units. For each class, there is one bar with height equal to thefrequency for that interval. This gives us a visual representation of the number of data values ineach class. The advantage of a histogram is that, with just a quick glance, we see how the data isdistributed. A histogram of our primary school completion data is shown in Figure 7. Notice thateach class is labelled with the lower bound for that class. So, for example, the bar with the 20underneath represents the three countries between 20% and 30%, as noted above in the frequencydistribution.17

Compared to the frequency chart, the histogram allows us to see patterns in the data moreeasily and quickly. For example, we easily see that more than half of the countries have completionrates more than 50%. In fact, by using the vertical scale, we can see there are exactly 7 that havecompletion rates below 50%, as we noted from the table. We also notice that there are no countrieswith rates between 10% and 20%, and 40% and 50%.

There are some standard terms that are used to describe histograms, and the examples belowwill illustrate these.

Figure 8 shows two histograms that are skewed right. This means that the peak is towardsthe left and the data trail off towards the right. As we will see later, data that are skewed rightwill have an average (mean) larger than the median.

The first histogram shows the distribution of child mortality rates for children under 5. Thedata are given in number of deaths per 1000 people in the population. We see that many countrieshave mortality rates on the low end, less than 50 deaths per 1000. The number of countries dropsoff drastically at first, and then shows an overall slow decease as the mortality rate increases.

The second histogram shows the distribution of faculty salaries, for tenured and tenure-trackfaculty, at the University of Maryland, a large research university. There are about 1200 faculty

17How histograms appear can vary with the computer software used. One unfortunate aspect of Microsoft Excelis that the scale numbers on the horizontal axis are sometimes shown in the middle of the interval, rather than atthe left end, which would be more standard


Histogram

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Figure 7: A histogram for Primary School Completion Rates (see footnotes).

members in the data set. The frequency has a peak at salaries between $50,000 and $60,000.There is a gap in the histogram towards the right, showing that there are no salaries between$140,000 and $160,000. The few salaries over $160,000 would be considered outliers. Outliers aredata points that are, in some sense, “far away” from the rest of the data. We will define moreprecisely what “far away” means in Chapter Three.

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Figure 8: Two histograms that are skewed right

Figure 9 shows a histogram that is skewed (slightly) to the left. It shows the frequency forthe percent of children who were ever breast fed for 62 countries. We see, for example, that thereis one country where more than 99% of infants were breast fed at least once. Notice that the(horizontal) range for this data is only from 87% up to 100%. This is a fairly narrow range,much narrower than the range in percentages in our primary completion data. As noted in thegraph, only 62 countries are represented, and so we might not consider the data representative ofall countries, especially without knowing which countries are included and which are not.

Figure 10 shows two histograms which we would call roughly symmetric. The first depictsnational poverty rates, and the second shows per capita incomes for the 50 states for 2002. The“tails” on each side of the peak are roughly the same length and overall, the histograms have aboutthe same shape on either side of the peak.

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Histogram of NationalPoverty Rates

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Figure 10: Two histograms that are roughly symmetric

Deciding whether a histogram is skewed or symmetric is somewhat subjective. Notice alsothat some of the histograms have more “up and down” from bar to bar. We might say that thesehistograms have some irregularity. Our first histogram on primary school completion rates mightbe called irregular.

Finally, Figure 11 shows a histogram that is bimodal, which means it has two well-definedpeaks. Not all bimodal histograms have the peaks at the extremes. The data shown here are fromthe Polity IV Project, which has compiled data on the nature of governments for all countries overtime, and given each country a Democracy Score based on the openness of the political institutionsin that country.18 At least according to this measure, countries tend towards complete democracyor complete lack of democracy.

Figure 12 shows a second bimodal histogram. This graph gives the frequency of bear weightsin pounds for a sample for 143 bears.19 There is a smaller, secondary peak to the right of the firstpeak. Since the sample consisted of 44 females and 99 males, the different peaks might representthe average for the two different genders (can you tell which is which?).

So far, we have discussed skewness merely as a description of a histograms shape. As we willsee in the next chapter, however, the main importance of skewness is that it tells you somethingabout the relation between mean and median of a set of data, and between these and peak of a

18Polity IV website is www.cidcm.umd.edu/inscr/polity/index.htm19The data was collected by Dr. Gary Alt and is available in Fathom as a sample document.


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Figure 11: Democracy Scores: A bimodal histogram

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Figure 12: Bear Weights: Another bimodal histogram

histogram. For a histogram which is skewed to the right, the mean will in general be larger thanthe median, and the tendency will be for both the mean and median to be to the right of the peak.The opposite will occur if the histogram is skewed to the left.

The Gini Scale for Income Inequality

Today, there is a huge disparity between the technology, education, health care and agri-cultural methods that are available in the developed and developing world. The principalchallenge we face is to close that gap...The countries, businesses and individuals thatare on the right side of the divide have to think hard about what kind of world they wantus all to live in 20 years from now. Narrowing the gap benefits everyone, and we have

24

the means to do it. If we don’t, we will have missed an amazing opportunity.20

— Bill Gates

We conclude this chapter by introducing the Gini Coefficient, which is designed to measureinequality. In particular, we will use Gini coefficients to measure income inequality in particularcountries, regions, or populations. Here’s how it works. Consider Table 5.

Cumulative % of Pop. 10 20 30 40 50 60 70 80 90 100Total % of Income 1 2 4 7 14 25 49 58 78 100

Table 5: Percentage of cumulative income versus cumulative percentage of population

Table 5 shows, for example, that if you took the bottom 10% of the population by householdincome, this group earns 1% of all the income earned in this country. The bottom 20% of thepopulation earns 2% of the income, etc. Note that the bottom 50% of the population earns 14%of all the income earned in the country, which means the top half of the population earns 86% ofall the income. This seems to be a country with a lot of poor people and very unequal incomedistribution. The data above is graphed below in Figure 13. This type of a graph of CumulativeIncome vs Cumulative Population is called a Lorenz Graph or Lorenz Curve.

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Figure 13: Lorenz graph for a society with significant inequality in incomes

Table 6 shows another example of an income distribution table.

Table 6 depicts the ideal egalitarian society. Every household earns exactly the same income!!We can tell this since the bottom 10% of society earns exactly 10% of all the income, the bottom20% earns 20% of the income, etc. This means, for example, that the “second 10%” (those in thebottom 20% but not in the bottom 10%) must earn 10% of all the income, just like the bottom10%. The Lorenz curve for this line would be a straight line, going from (0%, 0%) to (100%, 100%).

In Lorenz graphs, the closer the curve gets to the bottom right corner, the more inequalityexists in the population under consideration.

Figure 14 shows graphs for both Tables 5 and 6 together on the same coordinate axes.

20from P. Hofheinz (2001, January 29). Gates on technology, AIDS and why Malthus was wrong. RetrievedOctober 3, 2007 from ZDNet Web site: http : //news.zdnet.com/2100− 959522− 527652.html


Cumulative % Total Incomeof Population Earned by Pop

10 1020 2030 3040 4050 5060 6070 7080 8090 90100 100

Table 6: Table for a completely egalitarian society

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It is reasonable to think that the bigger the area between the dashed line, representing totalequality, and the given Lorenz graph, the more unequal the society. We define the Gini Coefficientto be twice this area between the two graphs.

The smallest this area can be is zero. What is the largest it can be? Well, imagine a countrywhere one person earns all the income and everyone else earns nothing at all. What would theLorenz graph look like?

Note that the total area under the dashed line is the area of a triangle with base equal to 1and height equal to 1 (for 100%). So, the area of this triangle is 1/2 which would mean a GiniCoefficient of 1 ( remember the coefficient is defined as twice the area). This is the largest a GiniCoefficient can be. You can think of the 1 as representing 100% inequality.

Using Figure 14, we can make a rough estimate of the Gini Coefficient for this population. Alittle less than half of the triangular area under the line is between the line and the curve. Notethat the area of each grid square is 0.22 = 0.04. By adding up squares and estimating areas ofpartial squares, we get an estimate of the area between the line and the curve of about 0.23. Thiswould give a Gini Coefficient of about 0.46 or 46%. In fact, it happens the the Gini Coefficient forthe United States in recent years has been around 0.45.

Reading Questions for Representing Quantitative Information

1. Plot each of the given points on the grid provided. Label the point with its correspondingletter.

A.(−1,4) B.(0,−3) C.(1,−5)D.(−2,1) E.(5,2) F. (−2,0)

X - axis

Y - axis

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2. Consider Figure 2 in the reading, which depicts World Oil and Gas Production.

(a) What is the window for this graph?(b) The graph shows not only total world production, but also breaks it down into various

countries and regions. Although one has to do a little interpretation, you can estimatewhen these other countries and regions reach their peak productions. Based on thegraph, estimate when Europe, Russia, and Other have or will reach their peaks. Explainhow you know.

(c) From the graph, estimate what percentage of total oil production was produced by theMidEast in 1970 and explain how you know. Do the same for 2000 and 2050.

(d) The graph indicates that the U.S. oil supply will be essentially depleted by 2035 andRussia’s by 2050. From the graph, estimate which of these two countries will haveproduced the greater amount of oil. Explain how you can tell?

3. Consider Figure 3 in the reading, the graph on marijuana use.

(a) What is the window for this graph?(b) Are there any characteristics of good graphs that this graph violates?

26


4. If you were to graph each pair of variables listed below, which variable would most appropri-ately go on the horizontal axis and which one would go on the vertical axis?

(a) The amount of gasoline you put into a car and the cost of the gasoline.

(b) The time it takes a car to accelerate from 0 to 60 mph and the horsepower of the car’sengine.

(c) Day of the year and amount of daylight (i.e. time from sunrise to sunset).

(d) A child’s height and the child’s age.

(e) The year and the cost of tuition for a full-time student at your institution.

(f) The number of words in the vocabulary of an eleven year old and the number of hoursthe child’s parents have spent reading to him or her over his or her life.

(g) The spending per pupil in public school in a state and the high school graduation ratein that state.

5. Suppose you were to redraw the graph from Figure 1 of the text, changing the window to1952 ≤ Year ≤ 2002, 5500 ≤ Oil Production ≤ 10000. How would this change the impressioncreated by the graph?

6. If someone wants to graphically de-emphasize a sudden increase or decrease, they will some-times make the vertical range of the graph window extend well beyond the range of the data.For example, suppose a school superintendent used a graph to represent the change in studentstandardized test scores from 1990 to 2002. The average standardized test score was 85 in1990 and was 72 in 2002.

(a) Use a vertical scale from 0 to 90 and sketch a plausible graph showing the change inaverage test scores.

(b) Use a vertical scale from 70 to 85 and sketch a plausible graph showing the change instandardized test scores.

(c) Which graph gives a more noticeable decrease? Why?

7. How are a frequency distribution and a histogram similar? How are they different?

8. What are the advantages of displaying data in a frequency distribution?

9. What are the advantages of displaying data in a histogram?

10. Figure 11 in the text shows the histogram for Democracy Scores, as defined and computedby the Polity IV Project. It is an example of a bimodal histogram, and we commented in thetext that, based on this shape, countries tended either towards total democracy or total lackof democracy.

(a) Suppose that instead of a bimodal histogram, the histogram for Democracy Scores wasskewed to the right. What would this tell you about democracy in the world? Youmight draw an example histogram that is skewed right. In this case, remember that allscores are between 0 and 10.

(b) What could you say if the histogram was skewed to the left?

11. By hand, construct a histogram using the following data, which gives Gini Numbers forall European countries. Use classes of width two, starting with 20. Make a sketch of thehistogram.

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Country Gini No. Country Gini No. Country Gini No.Belarus 28.53 Italy 28.30 Romania 25.33Belgium 26.97 Latvia 26.98 Slovak Rep. 21.51Bulgaria 27.80 Lithuania 33.64 Slovenia 27.77

Czech Rep 23.05 Moldova 34.43 Russia 30.53Denmark 29.06 Netherlands 29.16 Sweden 28.88Estonia 37.22 Norway 28.91 UK (Britain) 32.35Finland 23.55 Poland 26.68 Ukraine 25.71Germany 26.00 Portugal 34.44 Yugoslavia 31.88Hungary 27.87

(a) Describe the shape of your histogram.(b) Based on your histogram, estimate that average percent inequality for European coun-

tries.(c) Write a few sentences describing what this histogram tells you about income inequality

in Europe.

12. Answer the following questions about histograms using the data given below.

1 1 2 2 4 5 5 8 9 12 12 12

(a) Construct a histogram by hand using three groups: 0 to 4 (including 4, since our standardpractice is to include the upper end, but not the lower end of each class), 4 to 8, and 8to 12.

(b) Construct a histogram using four groups: 0 to 3, 3 to 6, 6 to 9, 9 to 12. Make a sketchof the histogram.

(c) Why do your two histograms look different? What does this tell you about constructinghistograms with different groupings?

13. Consider the histogram in Figure 9 of this chapter, showing the percent of children who wereever breastfed for 62 countries.

(a) In how many countries were 95% or more of children breastfed?(b) In what percent of countries were 95% or more of children breastfed?(c) If you picked one of the 62 countries represented in the histogram at random, what is the

probability you would pick a country where 95% or more of the children are breastfed?(d) Does the histogram support the statement “In 95% of all countries, at least 9 out of 10

infants were breastfed.” Explain why or why not.

14. Why would a scatterplot of people’s heights versus their total arm spans have a positiveassociation?

15. For each of the following, determine if the two variables would have a positive association, anegative association, or no association. Briefly explain.

(a) size of a building and cost of heating or cooling(b) automobile’s age and gas mileage(c) size of a city and number of violent crimes(d) number of songs on a CD and cost of the CD(e) amount of time spent watching TV and amount of time spent studying in the library

16. If you were to graph each of the following on a scatterplot, which variable would go on thehorizontal axis or does it not matter? Explain your answer.


(a) size of a building and cost of heating or cooling(b) automobile’s weight and gas mileage(c) size of a city and number of violent crimes(d) car’s price and age(e) number of years of education past high school and height

17. The graph below represents the percentage of the population in a number of countries whohave access to improved drinking water supplies.21Access to Improved Drinking Water

0102030405060700 10 20 30 40 50 60 70 80 90Percent of Population

(a) What type of graph is this?(b) Approximately how many countries are represented in this graph?(c) In about how many countries does at least 80% of the population have access to improved

drinking water? What percent of all the countries represented is this?(d) How would you describe the skew of this graph?(e) How do you think this graph would have looked using data from 100 years ago? Explain

why you think so.

21Data collected from the United Nations Conference on Trade and Development website, http://stats.unctad.org

30

18. Use an appropriate graph from this section to display the following data. Make a sketch ofeach graph.

(a) This data set represents weights (in kilograms) of 20 young women.

52 75 63 60 63 65 47 62 35 5555 51 72 58 53 53 65 50 50 71

(b) The following table shows per capita state tax revenues, as well as the number of bach-elor’s degrees for each 1000 18 to 24 year olds in several midwestern states both for theyear 2000.22

Bach. DegreesPer Capita per 1000 18 to 24

State Tax Revenue Year OldsIllinois 1,835 45.7Iowa 1,772 62.7Kansas 1,803 53.3Minnesota 2,711 49.2Missouri 1,532 55.9Montana 1,564 59.1Nebraska 1,742 61.7North Dakota 1,826 66.7South Dakota 1,228 61.3Wisconsin 2,345 52.8Wyoming 1,952 36.0

19. Consider the histogram on child mortality rates given as part of Figure 8 in the text. Writea few sentences interpreting what this graph tells you about child morality rates. Be asthorough as you can.

22Data from U.S. Department of Education, National Center for Education Statistics, Integrated PostsecondaryEducation Data System, various years; and U.S. Bureau of the Census, Population Division and the U.S. CensusBureau


Representing Quantitative Information: Activities and ClassExercises

1. Wind Energy. Wind energy has become more prevalent in recent years. In South Dakota,wind turbines have been built in a number of areas.

(a) Open the Fathom file Wind Energy. You should see a collection and a case table. Thevariables, or attributes in the language of Fathom, are TotalMonths, Year, Month andEnergy. The TotalMonths counts the number of months with January, 1998 countingas 1. The Energy variable is measured in Trillions of BTU,23 and represents the totalwind energy produced in the U.S. each month over this period of time. Create a graphof Energy versus TotalMonths. Recall that you can do this by first dragging a graphinto the main window, and then dragging the appropriate columns to the vertical andhorizontal axes of the graph box.

(b) In a few sentences, describe what this graph tells you about wind energy production inthe U.S.

(c) Notice that the graph has somewhat of a wavy pattern of peaks and valleys, at leastafter the first year or so.

i. Investigate when (during which part of the year) these peaks seem to be happening.What time of year do the peaks seem to be happening?

ii. What are some possible explanations for the peaks happening at these times?

(d) Remove the TotalMonths variable from the graph by selecting Graph—Remove XAttribute. The graph should change to a dot plot, which shows one dot for eachmonth, and places that dot along the y-axis at the Energy value for that month.

i. Use the Object menu to make a duplicate copy of the dotplot. You should nowhave two dotplots in the window.

ii. Change one of the plots to an Energy versus Year plot. Write a couple of sentencesdescribing what this graph tells you about wind energy production.

(e) Change the dotplot that is still on the screen to a histogram (click on the arrows in thebox at the upper right of the Graph Box).

i. Move the mouse over a boundary between two bars until you get a double-arrow.Then click, hold, and drag. Describe what happens.

ii. Experiment a little with changing the size of the bars. Move the boundaries arounduntil you get each bar to cover 4 units on the vertical axis with the first bar startingat 2. You should get a total of 4 bars (one for 2 to 6, one for 6 to 10, etc.).

iii. Once you get 4 bars in your histogram, some may go out of the window to the right.To rescale the graph so you can see everything, move the mouse over the horizontalscale values on towards the right of the axis until you get a “grabbing hand.” Then,click hold and drag. Play around with this until you see the whole histogram. Trymoving the hand to different parts of the the horizontal axis and also the verticalaxis, and then drag to see what happens.

(f) Be sure to save your file onto your H drive or some location where you will be able tofind it later.

23BTU stands for British Thermal Unit, and is a standard way of measuring energy. Eight gallons of gasolinecontains about 1 million BTU.

32

2. Hungry Planet. In preparing their book Hungry Planet, authors Peter Menzel and FaithD’Aluisio visited 30 families around the world and documented their entire food intake andits costs for a week. The histogram below shows the frequency for the cost per person forthese 30 families. Note that the horizontal axis scale is offset; the first bar indicates thatthere were 12 families where the cost of food per person for a week was between $0 and $20.

02468101214

0 20 40 60 80 100 120 140Frequency

Dollars

Weekly Cost Per Person

(a) What skew does this histogram display?(b) In how many families was the weekly cost per person more than $60?(c) In what percentage of these families was the weekly cost of food per person less than

$40?(d) For each member of your group, estimate as carefully as you can what you spend on

food per week. If you are on a meal plan, include your weekly cost for the meal plan(even though this expense would include service, clean up, etc. that are not really partof the figures in Hungry Planet). Be sure to explain how you came up with the estimatesfor each person in the group.

3. Periodical Search. Using a newspaper, magazine, or other periodical do the following:

(a) Each group member should find at least one example of quantitative information ingraphical form.

(b) For each graph:i. Include a copy of the graph (scanned if possible).ii. Indicate the source of the graph (e.g. the July 3rd edition of the New York Times).iii. Indicate the type of graph (e.g. scatterplot).iv. Indicate what the graph represented. Include any variables named.v. Indicate the window used for any two-variable graphs. Indicate the scale for other

types of graphs as appropriate.vi. Write a few sentences on what the graph is telling about the situation depicted.

Consult the text for examples.(c) Did any of the graphs violate good graphing practice? Explain briefly.


4. Anscombe I. The graphs below are scatter plots of 4 different two-variable data sets pro-duced by Anscombe24 as examples related to linear regression, which is a topic we will beconsidering later in the course. In this activity, you will “get acquainted” with the data setfrom a visual standpoint. We have labeled the 4 plots A, B, C, and D to the lower left ofeach plot.

A.

456789

101112

B

4 6 8 10 12 14 16 18A

Anscombe Examples Scatter Plot

B.

3456789

10

C

4 6 8 10 12 14 16 18A


C.

56789

10111213

D

4 6 8 10 12 14 16A


D.

56789

10111213

F

8 10 12 14 16 18 20E


(a) For each of the 4 graphs, make your best prediction of the y-value if the x-value was 16,assuming the visual trend continues.

(b) Consider the predictions you made. Based on what you see in the graphs, you may bemore or less confident about the accuracy of these predictions. In other words, if therewas additional data which included y-values for x = 16, you might be very confident ornot very confident that your prediction would be close to the actual y-value. For whichgraphs do you feel your prediction is highly likely to be accurate? Which predictions doyou feel have a good chance of being inaccurate?

24Data from Anscombe, F.J., Graphs in Statistical Analysis, American Statistican, 27, 17-21

34

5. Gini Coefficients and Lorenz Curves. This activity is related to Lorenz Curves and theGini Coefficient which were discussed in this chapter. Review the relevant sections as neededto do this activity.

(a) Draw the Lorenz curve, and calculate the Gini coefficient for a hypothetical countrywhere the bottom half of the population earns nothing at all, and everyone in the tophalf earns exactly the same amount.

(b) Draw the Lorenz curve, and calculate the Gini coefficient for a hypothetical countrywhere one person earns 50% of all the income and everyone else earns exactly the sameamount.

(c) Draw the Lorenz curve, and calculate an estimated Gini coefficient for a hypotheticalcountry where the bottom 90% of the population earns a total of 10% of all the income.

(d) Draw the Lorenz curve, and calculate the Gini coefficient for a hypothetical countrywhere the bottom half of the population earns exactly the same amount, and the tophalf also each earn the same amount, but each person in the top half earns twice asmuch as each person in the bottom half.

(e) Think about a hypothetical country with a population of 10 and only two small busi-nesses. Suppose the occupations of these 10 people are unemployed, two fast food work-ers, a fast food restaurant manager, a janitor, a groundskeeper, a ruler, two computerprogrammers, and an accountant.

i. Assign incomes to each of the 10 people and put them in increasing order. Puttingthem in a table might be helpful.

ii. Draw a Lorenz curve for this country based on your assigned incomes.iii. Calculate the Gini Coefficient for this hypothetical country.

6. Income Inequality in Three Countries. The graph below shows the Lorenz curves forBangladesh, Brazil, and the United States for the year 1989.

Sample Lorenz Curves

0.000.100.200.300.400.500.600.700.800.901.00

0 0.2 0.4 0.6 0.8 1

Bangladesh

Brazil

USA

Equality

(a) Give the percentage of total income earned by the bottom half of the population in eachof the three countries.


(b) Give the percentage of total income earned by the top 20% of the population in each ofthe three countries.

(c) Calculate an estimated Gini coefficient for each country, based on the graphs. As notedin the text, the Gini coefficient is twice the area between the equality line and the graph.

(d) Of the three countries for which data is given, Brazil has the highest level of inequalityand Bangladesh the least. However, one drawback of the Gini coefficient is that itdoes not take into account the overall level of income. Per capita incomes around thistime for Bangladesh, Brazil, and the U.S. were approximately $150, $2500, and $17,000respectively. The populations for these countries (in millions) were approximately 109,148, and 255.

i. Based on the additional data, find the total amount of income earned in each coun-try.

ii. Using the Lorenz curves, find the total amount earned by the bottom 20% of thepopulation in each country. About how much is this per person?

iii. Find the total amount earned by the top 20% of the population in each country.About how much is this per person?

7. Looking at Democracy Vs Illiteracy.

Consider the following quote by Gwynne Dyer, a former Canadian statesman.

Take the year a country first reaches 50% literacy, and add one or two generationsto allow the idea to sink in, and, democracy, more or less automatically, appears.— Gwynne Dyer

Is Dyer correct?

Consider the scatter plot below which shows the literacy rate in 1970 as the independentvariable and the Polity IV project’s Democracy Score for the year 2002 as the dependentvariable. The literacy rate is the percentage of the population 15 years of age or older whocan read well enough to be functionally literate.

Democracy 2002 vs Literacy 1970

0

2

4

6

8

10

12

0 20 40 60 80 100

Literacy Rate as Percent of Pop over 15

Dem

ocr

acy

Sco

re

36

(a) Does the scatterplot seem to have a positive association, negative association, or noassociation? Keep in mind that areas where points are more densely clustered shouldbe given greater weight in deciding what the overall trend is.

(b) Does this scatterplot tend to confirm or to refute the Dyer quote? Explain as completelyas you can what the plot seems to imply.

(c) There are quite a number of countries with a Democracy Score of 0. Although most ofthese have low literacy rates, a few have higher literacy rates.

i. Speculate a little bit about which countries or regions of the world would have lowDemocracy Scores but high literacy rates. Include a list of at least a half dozencountries.

ii. Using the tables in the Data Samples Appendix, list which countries have a Democ-racy Score of 0 but literacy rates over 40%. How close to reality were your specu-lations?

iii. Give some possible explanations or hypotheses about why these countries have lowDemocracy but high literacy.

8. Income Inequality in the United States

The Gini Coefficient is one way to measure inequality. Another way is to compare incomesfor different income groups within a population. The graph below shows Percent Increasein Real Average Income as a function of Time for six different income groups in the UnitedStates. Real income means income adjusted for inflation. In other words, if your incomegoes up at exactly the rate of inflation, your real income stays exactly the same. So, if thegraph of real income versus time were a horizontal line, this would mean that real incomehas neither increased nor decreased, which in turn means that the actual income would havegone up at exactly the same rate as inflation.25

Each line in the graph represents how average real income has changed for the given group.For example, the graph for the “1st fifth” represents the percentage change in average incomeof those earning incomes in the bottom fifth of all incomes. These would be the poorest wageearners. Graphs are given for each fifth, as well as the top five percent, which would be therichest five percent. This richest five percent is a part of the top fifth.

The values for the dependent variable were calculated by taking the average income for thegiven group in the given year, and dividing by the average income in that group for 1975.For example, the average real income for those in the top five percent in 1975 was $120,364and in 1994 it was $183,044. Taking 183, 044/120, 364 we get 1.52, which is the value shownin the graph for 1994 for the top five percent group. This means that average real incomewent up 52% from 1975 to 1994 for this group. For comparison, the average real incomes forthe bottom fifth in 1975 and 1994 were $8001 and $7762 respectively.

(a) Which group experienced the largest percentage increase in real income over this 19 yearperiod? Which group fared the worst with respect to change in real income?

(b) Which groups, if any, had their actual income increase more slowly than inflation from1975 to 1994?

(c) Why do all six plots have a y-value of 1 in 1975?

(d) What overall picture does the graph give regarding changes in income in the U.S. forthis period?

(e) Suppose you redrew the graph, changing the range on the y-axis from .9 up to 1.6, to 0up to 2.0. How would this change the impression of the graph?

25The details, which are not that hard, of calculating values which are adjusted for inflation will be covered inChapter 7

2. Representing Quantitative Information 37Percent Increase in Average Real Income by Income Group: 1975-19940.911.11.21.31.41.51.6

1975 1980 1985 1990 19951st Fifth2nd Fifth3rd Fifth4th FifthTop FifthTop 5%

(f) Could you change the impression of the graph by changing which years you included?For example, what would happen if you started in 1985 or 1990 instead of 1975? Do youthink the person who created this graph could have manipulated the impression createdby his selection of years? To help answer this question, here is a graph of percentageincreases in income for the lowest fifth and second lowest fifth along with the top fivepercent starting in 1967.26Percentage Increases in Real IncomeAverage Income for Each Group: 1965-19940.911.11.21.31.41.51.61965 1970 1975 1980 1985 1990 1995 1st Fifth2nd FifthTop 5%

Figure 15: Another graph of percentage increase in real income

(g) Both of the graphs above depict percentage increases in average real income, rather thanaverage real income itself. On the same set of axes, create the best plots you can for theactual average real income for both the top 5% group and the bottom fifth, starting inthe year 1975. The dependent variable would be measured in dollars. Explain how youcreated your graph. How does this change the impression created by the graph?

26Note the significant increase for the bottom fifth from 1965 to 1974. This might be at least partially attributedto the “War on Poverty” initiated by the administration of President Johnson.

38

9. Bowling Alone. In his book, Bowling Alone: The Collapse and Revival of American Com-munity, author Robert Putnam defines what he calls the Social Capital Index, which is ameasure of how “community minded” people are. The higher the Social Capital Index for agroup of people, the more likely they are to interact in constructive ways, join clubs, partic-ipate in community activities, etc. Putnam has computed the Social Capital Index for eachstate, except Alaska and Hawaii.The first scatterplot in Figure A below shows the relationship between the Social CapitalIndex (SCI for short) and Percent Food Insecure Households (PFIH for short). PFIH rep-resents the percent of households in a state that typically have trouble procuring sufficientfood, or food of adequate quality on a monthly basis. It is considered a measure of how manypeople are at risk for hunger.The second scatterplot shows the relationship between the SCI and State Poverty Rates,defined as the percentage of the population in a state that are living below the official povertyline. Data in tabular form for these and other variables are given in the Appendices.Figure C shows the relationship between Educational Spending per Pupil and the SCI.

Percent Food Insecure Households

5

7

9

11

13

15

17

-2 -1 0 1 2

Social Capital Index

Figure A

(a) For each scatter plot, say whether the variables seem to have a positive association,negative association, or no association.

(b) Which relationship seems to be the stronger, or are they of about the same strength?(c) In the poverty versus SCI plot, the point at the lower left could be considered an outlier.

It is the lowest of all states in Social Capital, and is also has a fairly low poverty rate.Using the table in the Appendices, find out which state this is.

i. Can you think of some reasons why this state is low in Social Capital?.ii. Can you think of some reasons why this state has a fairly low poverty level?

(d) In the Education Spending Versus Social Capital plot, the state that is lowest in perpupil spending has an SCI of .5. Find out which state this is.

i. Can you think of any reasons why this state would be low in per pupil spending,but in the middle with regards to social capital?


State Poverty Rates

5

10

15

20

25

30

-2 -1 0 1 2


Figure B

Spending Per Pupil 1999-2000

0

2,000

4,000

6,000

8,000

10,000

12,000

-2 -1 0 1 2


Figure C

3. Collecting and Analyzing Data

In this chapter, we make a more systematic study of some of the basic concepts that are usefulwhen considering real world data. The basic theme will always be to ask questions of the data.What can I learn from this data? Are there any conclusions I can draw? Can I verbally describethe situation under consideration based on the data I have?

It is also important to be cautious when working with data. Can I trust that this data setis reliable? What are the shortcomings of this data set? Are there any important cases or otherinformation that seem to be absent from these data? Overall, we want to strike a balance betweenanalyzing and understanding the data as completely as possible, but without jumping to conclusionsby reading more into the data than is actually there.

Validity and Reliability

It has been widely reported by a number of media outlets that the average age of a homelessperson is nine.27 Can this number be right? Where did it come from?

The first consideration when working with data is knowing to what extent the data are validand reliable. Validity refers to how well the data measures what it is intended to measure. If yourbathroom scale gives the true weight of the people that stand on it, the weights it produces forma valid set of data. If your bathroom scale does not produce accurate weights, or if it gives theheight instead of the weight, then it is not producing valid data. Valid measurement techniquesare also sometimes referred to as accurate.

The reason validity is important is that if we answer questions, conduct research, make predic-tions, or base public policy on invalid data, then our answers, research conclusions, predictions, andpolicies could well be wrong or counterproductive. If we develop policies to address homelessnessassuming that the average age of a homeless person is nine, when the average is really 39, ourpolicies are likely to be ineffective or wasteful.

Reliability refers to a data collection procedure producing consistent results in repeated use.If your bathroom scale consistently gives weights which are 10% heavier than the actual weight,then your scale produces reliable data, but not valid data. Another way of putting this is that ifyou weigh yourself on the same scale at different times or in different places (while still actuallyweighing the same amount), then your scale produces reliable data (even if the data is not valid!).

In many situations, validity and reliability are not a problem due to the nature of the data.For example, if we are collecting data on a certain sport, say baseball, we can easily access datathat is entirely valid and reliable for each individual or each team. Major League Baseball’s officialrecords would have the exact number of wins and losses for each team, the exact number of homeruns and stolen bases for each players, etc. Errors in measuring these statistics are very unlikely.The biggest possibility for error is probably in typing or copying the data incorrectly.

In other situations, validity or reliability can be a problem. For example, we might want tocompare household incomes for the 50 United States, or among various countries. How wouldwe determine how much individual households make in a given state or country? It is typicallymuch too impractical to ask every household to provide us with their income. We typically mustbe satisfied with collecting data from only a sample of the entire population of all households.In general, a population is the entire group we would like to have some data on, and a sampleis a subset of the population from which we actually collect data. The key to obtaining validand reliable data in this situation is using a sampling technique that will consistently producerepresentative samples of the population.

For example, if a high school student wanted to collect data on household incomes, she mightsimply to decide to knock on 100 different doors in her small Nebraska town and ask one of theresidents what their household’s income is. If she is careful to randomly select which houses tovisit, her data might be fairly representative of household incomes in her town, or in other words,

27see http://abclocal.go.com/wjrt/story?section=news/local&id=4975476 for one example, or simply google thelast nine words of this sentence

40

3. Collecting and Analyzing Data 41

if the population she wanted to consider was all households in her town. However, her data wouldnot be representative if the population under consideration was all households nationwide, sinceno urban households and no households outside of Nebraska were included in her survey.

So, if the student were using this technique to estimate incomes nationwide, her method wouldnot be valid. It would not accurately measure incomes nationwide. On the other hand, if sherepeated her survey several times, each time randomly selecting 100 households, the method wouldprobably be fairly reliable. The averages of her several samples would likely not vary from eachother too much. How much variance she might expect we will be able to quantify when we discussstandard deviation.

It is also possible the people in an individual household do not have a very accurate idea ofwhat their annual income is or might have some motivation to be less than truthful in reportingtheir income. In cases like this, even a representative sample might not produce valid data. Incollecting data, or using data that has been collected for us, we always want to be aware of potentialproblems in the validity and reliability of the data.Example 1. A high school student wants to determine what percentage of students in her school

have engaged in underage drinking. To this end, she randomly selects 25 of her schoolmates andschedules face-to-face interviews with them in an empty classroom. Interviews take place duringschool hours. One question asked during the interview is ”how many times have you engaged indrinking alcohol in the last six months?” From the 25 responses she receives, she finds that theaverage number of times the respondents drank was 1.3.

Does this method produce a valid and reliable average?

Solution:Given the circumstances of the interviews, the validity of the data is questionable. The students

interviewed may be reluctant to be totally truthful, given the possibility that their responses mightbe overheard by school officials or otherwise be made available to persons other than the studentdoing the interviews.

As far as the reliability, this may be difficult to determine. If the student conducted severalsets of 25 interviews with different groups of randomly selected students, she probably will notget the same 1.3 average every time. On the other hand, we might not expect the results tobe drastically different from group to group. Later in this chapter, we will be able to quantifyhow much ”variability” we can expect in a situation like this. In general, if the student employsappropriate randomization and asks a sufficient number of students, we would consider the datareliable.

Collecting a Sample

Random Samples

Data can be collected in quite a number of ways. As noted above, data are often collectedthrough surveys or other sampling techniques. In this section, we discuss issues related to sampling.

Collecting a proper sample is key to producing good data. Without it, the conclusions that youdraw from your sample are usually worthless. Typically, national polls are conducted by samplingapproximately 1000 people since this gives a margin of error of no more than three percent.28 Fromthis relatively small number, the opinions of more than 300 million Americans can be inferred witha relatively high level of confidence. We call the percentage of all 300 million Americans whorespond in a certain way to a survey question a population parameter p. The percent of thesample who respond in this way is called the sample proportion, and is denoted p̂.

The most important aspect in collecting data is randomization. In order to create a samplewhich is representative of the population from which it is drawn, we seek to use methods whichwill insure (as much as humanly possible) that each member of the population has an equally likelychance of being in the sample. The most basic sampling technique which accomplishes this is calleda simple random sample.

28We will see how margins of error are calculated and what they mean in the next chapter

42

For example, suppose we want to find out whether the students at a small college of 1000students object to a new policy banning the sale of soft drinks on campus. To create a simplerandom sample of size n = 20 from this population of 1000 college students, we could assign eachstudent a number from 1 to 1000, write these on 1000 identical pieces of paper, put them all ina hat and then draw out 20 (after mixing thoroughly). This should give each student an equallylikely chance of being selected, and as a result, give each possible set of 20 students an equallylikely chance of being the sample.

In practice, one uses more technologically sophisticated mechanisms for achieving randomness.Many computer programs have so-called “random number generators” which can be used insteadof a hat full of paper to do sample selection.

Non-random sampling techniques can create invalid, often also called biased data. For example,if we were doing our soft drink survey but selected our sample of 20 by standing next to a cokemachine in the basement of a residence hall and asking the first 20 students who wander by, weare not giving each student an equally likely chance of being selected. We would tend to get onlystudents from that residence hall, and probably get students who like to drink pop, since we werestanding by a coke machine (and one located in an out of the way place at that). Thus, our resultsare not likely to be representative, and we can be fairly sure the people in the sample wouldhave more concern about the new policy than the student body as a whole. Our sample wouldbe considered biased, and as a result, the data from our sample are not likely to be valid. Wewanted an accurate measure of what percent of all students objected to the new policy, and oursampling technique is not likely to provide this.

There are other types of sampling techniques which can also produce results which effectivelyrepresent the entire population. One of these is called a stratified random sample. In a stratifiedrandom sample, the population is first divided into subgroups and then simple random samplesare taken from each group. For example, if we conducted our soft drink policy survey by firstdividing the 1000 students into residential and non-residential and then creating separate randomsamples from each group, this would be a stratified random sample. The advantage to such asample is that we could then see whether the two groups differed in their opinions and by howmuch.

Bias and Variability

In any survey, randomization is used to reduce bias. In general, any sampling technique whichis likely to produce data which is not valid is considered biased. Bias often occurs when the sampleselected is not representative of the entire population.

Another goal in sampling is to reduce variability. A sampling technique has high variability ifthe results obtained vary significantly in repeated use of the technique. In other words, if there issignificant variability, the technique does not produce reliable data. These two ideas are illustratedin Figure 1. The center of the target represents the parameter, p, for the entire population and thedots are the different p̂ determined by several different samples. If variability is low, the dots willbe clustered together. If bias is low, the dots will be around the center. Note that one can havehigh bias and low variability, or low bias and high variability. For example, if we conduct severalsoft drink surveys, but always stand near coke machines, the surveys might have low variabilitybut will probably have high bias. We are likely to consistently overestimate the percent of studentsp who object to the new policy.

If a sample is selected randomly, the possibility of getting a biased sample that systematicallyfavors a particular outcome is greatly reduced. There are, however, other things that can bias astudy. For example, the way in which questions are worded can bias the results. For example,asking the question “Do you favor allowing students the freedom to choose to drink soft drinkswherever and whenever they wish?” is likely to get more “ yes” responses than the question “Doyou favor exposing students to risks associated with obesity by allowing them unlimited access tosoft drinks?” These statements admittedly are very different in wording, but even subtle changesin wording questions, such as changing “welfare” to “aid to the poor”, or “pro-choice” to “pro-abortion” can influence the responses people give to a question.

The principle way to reduce variability is to use larger sample sizes. In general, the more people


Low bias and low variability Low bias and high variability

High bias and low variability High bias and high variability

Figure 1: An illustration showing varying combinations of variability and bias.

surveyed, the less variability. Large does not mean hundreds of thousands, however. Political polls,for example, typically sample about 1000 people. The margin of error usually reported for asimple random sample of 1000 is about 3%. This typically means that we are very sure (where“very” usually means 95% sure) that our poll result is within 3% of the population parameter pwe are trying to measure. If the sample were to be reduced to 100 instead of 1000, the margin oferror would increase to plus or minus 10% which is too high to be acceptable for most studies.

To summarize, bias is generally controlled by employing randomization and careful design ofquestions, while variability is generally controlled via sample size.

Example 2.

The soft drink survey above is one example of a poor sampling method. Here is another.Each day, many news websites like CNN conduct surveys. On July 20th, 2007 the question

on CNN’s main page was “Do you plan to read the new Harry Potter book?”29 One could selectYes or No. This sampling technique is certainly not random. Only those who go to the CNN sitewill even see the poll. Of those who see it, those who have strong feelings (good or bad) aboutHarry Potter are more likely to respond than those who could care less. Clearly, not every personin the U.S. is equally likely to be a part of this sample. Samples like this are called self-selectedor also voluntary response samples. If you are curious about the results, as of 6 p.m. CentralDaylight Time there were 15458 yes votes, 30% of the total, and 36822 no votes. Also, CNN didindicate on their website that this was not a scientific poll. In other words, they acknowledgedthat randomization was not used and that the results were not necessarily representative.

29Harry Potter and the Deathly Hallows, the last in the Potter series, was released the following day with over 2million advance sales in the U.S.

44

When you read articles that include results from sampling, be careful not to “believe everythingyou read.” Ask questions. Did the sampling technique employed use randomness or was the samplecompletely self-selected? What was the sample size and the margin of error? How were thequestions worded? Was the survey or study conducted by a reputable firm with experience in suchwork, or was it conducted by an organization likely to have a strong view on the issue at hand?This latter does not automatically mean the results are biased, but it should lead one to be carefulin accepting the results at face value.

It is important to note that in this chapter, we use the word “random” in a very specific well-defined way. On the other hand, “random” also has other connotations that are less specific. Forexample, we speak of events happening “at random” like a “random accident.” Here, “random”means something like “unexpected” or “unable to be anticipated.” Or, a student might decide todo a survey by asking people he runs into “at random” her set of questions. Again, “random” heremight mean “not pre-determined.” However, this is not a random sample in the mathematicalsense because the student has not insured that each person in the designated population is equallylikely to be part of the sample.

Measuring Center

Average real family income grew by well over 15% from 1982 to 1989.—Rush Limbaugh

For millions of years, on average, one species became extinct every century. ...We arenow heaving more than a thousand different species of animals and plants off the planetevery year.— Douglas Adams

The average age of the world’s greatest democratic nations has been 200 years. Eachhas been through the following sequence: From bondage to spiritual faith. From faithto great courage. From courage to liberty. From liberty to abundance. From abundanceto complacency. From complacency to selfishness. From selfishness to apathy. Fromapathy to dependency. And from dependency back again into bondage.— Lord Thomas MacCauley

In this era of globalization, international affairs touch the lives of average Americansin unprecedented ways. And as we wage a global campaign to purge from the world,the terrorist threat against our very way of life, the assistance we provide to friendlygovernments and impoverished peoples across the globe supports our ability to sustainan international coalition to fight terror and retain the popular goodwill necessary tothis task.— Senator John McCain

The quotes above illustrate a number of different ways that the word “average” is used. Whenwe are considering a set of data, the average is usually meant to be the single number which bestrepresents all the numbers in the data set.

Mathematicians refer to averages as “measures of center,” and there are actually several waysto do this, including the common average that everyone is familiar with. In this section, we willlook at two ways of measuring the center of a data set. We will also consider the concept of thespread of a data set and ways to numerically describe this spread.

Intuitively, the center of a set of data is a single data point or value that is meant to be mostrepresentative of the whole set. Obviously when you represent a whole set of numbers by a singlenumber, you are going to lose some information. No one number can accurately represent thewhole set.


The Mean of a Data Set

There are several measures of center that are commonly used. The first is the ordinary arith-metic average, also known as the mean. To determine the mean of a data set, you add up the allthe data values and divide by the number of data points. For example, suppose the starters on acollege basketball team have heights in inches of 72, 76, 77, 77, and 80. The sum of these heightsis 382 and the mean of these numbers is (72 + 76 + 77 + 77 + 80)/5 = 382/5 = 76.4 inches. Thestandard symbol for the mean of a data set represented by x is x, which is usually pronounced“x-bar.” For our starting team, we have x = 76.4 inches.

You can think of the mean as a way of “levelling off” a data set. For example, our basketballteam above would have the same “total height” of 382 inches as a team of 5 players all of whomwere exactly 76.4 inches tall. Thus, the mean is always the value of each data point if the totalsum of all the data points is divided equally.

As mathematicians always like to be efficient, the following notation is often used as shorthandwhen dealing with means. If there are n numbers, denoted by the generic names x1, x2, x3, . . . , xn,their mean is

x =x1 + x2 + x3 + . . . + xn

n.

This can also be expressed as

x =

n∑

i=1

xi

n.

The Σ is the upper case Greek letter sigma, and is the standard symbol for summation. Theexpression, Σxi, is just a symbolic way of saying “add up the numbers” x1 + x2 + x3 + . . . + xn.

Medians

A second common way of measuring the center of a set of data is the median. The medianis the middle number of an ordered set of data. To find the median height for our basketball team,we first order the data set as {72, 76, 77, 77, 80} (or put in reverse order). The middle number, 77,is the median.

What if there is not middle number? In this case, the median is the average of the middle twonumbers in the ordered set. For example, to determine the median for the data set {7, 5, 9, 11, 6, 2},we first order the numbers from smallest to largest as we did before: 2, 5, 6, 7, 9, 11. The two middlenumbers are 6 and 7, and so the median is the average of these two, namely 6.5. The median of adata set has the property that the number of data points larger than the median is equal to thenumber of data points smaller than the median. This is why the median is a measure of center forthe data set.

Example 3. The following table shows malnutrition data for a number of Latin Americancountries for the year 2001. The data is the percent of children under five who are severelyunderweight for their age. 30

Determine the mean and median for this data.

Solution: The mean percentage of severely underweight children is

x =0.4 + 0.6 + 0.7 + 0.8 + 0.8 + 1 + 1.1 + 1.2 + 1.7 + 1.9 + 1.9 + 4 + 4.7

13= 1.6.

This means that, on average, these countries have a 1.6% rate of malnutrition, using this measure.To determine the median, note that the data is already listed from lowest percentage to highest.Since this is an odd numbered data set, the median is just the middle number, which is 1.1. Thistells us that approximately half of the countries have underweight rates higher than 1.1%, and halflower.

30Data is from the UNICEF end of decade database at <http://www.childinfo.org/eddb/malnutrition/index.htm>.

46

Country % Severely Underwt. Chld.Costa Rica 0.4

Brazil 0.6Venezuela, RB 0.7

Colombia 0.8El Salvador 0.8

Dominican Republic 1Peru 1.1

Mexico 1.2Bolivia 1.7Ecuador 1.9

Nicaragua 1.9Honduras 4Guatemala 4.7

It is an unfortunate fact that both the mean and the median are often referred to as theaverage. In considering the data in Example 2, one newspaper might say “the average LatinAmerican country has a malnutrition rate of 1.1%”, while another might report this average as1.6%. Both deliberate and unintentional confusion of mean and median is common in political andpublic discourse.31 Whenever you hear an average reported, you should wonder whether you aregetting the mean or the median. Is there any way you can tell without actually going back to thesource of the data? Not always, but we will look at some ways where you might at least be ableto tell which is more likely being reported.

Comparing the Mean and Median

Although the mean and median are different measures of center, they often give similar results.For example, we found earlier that the mean of our five basketball players heights {72, 76, 77, 77, 80}is 76.4 while the median is 77. However, there are times when the median and mean are quitedifferent. In Example 2, we saw that the mean percentage of severely underweight children in thegiven countries is 1.6 while the median number was 1.1. Here, the mean is significantly largerthan the median, as compared to the range of percentages for all the countries. The reason themean and median are noticeably different is because of the impact of Guatemala and Honduras.The data for these two countries is significantly higher than the rest, and they could be consideredoutliers. 32

Intuitively, an outlier is a data point that falls well outside of the overall pattern of the data.When calculating the mean, every number is added to the sum. Thus, larger values like those forHonduras and Guatemala can make the sum, and thus the mean, significantly larger than it wouldbe if these data points were only slightly higher than the others. The median, however, is notimpacted by outliers because its calculation is based on just the middle number, or the middle twonumbers in the list. In cases where there are outliers or the distribution is severely skewed, themedian probably represents a better measure of center than the mean.

When a distribution is symmetric, such as in Figure 2, the mean and median both occur nearthe center or peak of the histogram and will be approximately equal. The median always occursat the point that divides the histogram into two parts of equal area. We will represent the medianin a histogram with the letter M . Equivalently, the sum of the heights of the bars to the left of Mshould roughly equal the sum of the heights of the bars to the right of M , as long as all the barsare of equal width.

31You might consider the Limbaugh quote opening this chapter regarding whether he is giving the mean or themedian, and whether he is doing this intentionally or not!

32The key here is what we mean by “significantly.” This will be quantified more precisely later.


The mean would be the “balancing point” of the histogram. In other words, if you imagineyour histogram bars were made out of blocks and you set them on a see-saw, the point at whichyou would you put the fulcrum of the see-saw to exactly balance it would be the mean. Just as ona real see-saw, the further you sit away from the fulcrum, the more you pull down on your end ofthe see-saw. We all know a small person far away from the fulcrum can balance a large person whois close to the fulcrum. In a histogram, a little bit of data (short bars) far away from the fulcrumcan balance a lot of data (tall bars) close to the fulcrum. For the distribution shown in Figure 2,both the mean and median are 18, and are indicated under the horizontal axis on the graph.

01234567

3 6 9 12 15 18 21 24 27M

ore

M_x

Fre

qu

ency

Figure 2: The mean and median are equal when the distribution is symmetric. In this distribution,they are both 18.

How do the mean and median compare if a distribution is not symmetric? Consider the dataon drinking water from the Reading Questions in Chapter Two. The graph in Figure 3 representsthe percentage of the population in a number of countries who have access to improved drinkingwater supplies.33

This histogram is severely skewed to the left. To estimate the median, one could first add upthe estimated heights of all the bars (it turns out this is equal to 150 which is the total number ofcountries represented). Half of 150 is 75. To find the median, we need to find that point M where75 or half of the countries have a percentage of the population with access to improved drinkingwater above M and half below. We see that around 60 countries have a percentage above 90, andabout 85 countries have a percentage above 80. This means the median M must be somewherebetween 80 and 90. Based on the actual data, the median turns out to be 85.

Estimating the mean is more difficult, but the mean will clearly be smaller than the mediansince the short bars far to the left of the peak will “pull the average down.” The mean calculatedfrom the actual data turns out to be 79.22.

Finally, Figure 4 shows a histogram of our Latin American malnutrition data. The histogramcould be considered to be skewed right. Alternatively, one could consider the two countries repre-sented by the rightmost bars as outliers. Either of these is consistent with the mean of 1.6 beinglarger than the median of 1.1. You might pencil in marks for these values on the histogram. Notethat, in situations where you have two competing numbers, both claiming to be the “average,”you might be able to tell which is the mean and which is the median if you have some idea of the(likely) skew of the distribution of the data.

In general, how do you decide if the mean or the median is a better measure of center? Theanswer depends on what you are looking for and how the data is distributed. If the distribution isapproximately symmetric, the mean and the median will be approximately the same. If the data is

33Data collected from the United Nations Conference on Trade and Development website, http://stats.unctad.org

48 Access to Improved Drinking Water010203040506070

0 10 20 30 40 50 60 70 80 90Percent of PopulationFigure 3: Histogram of data on improved drinking water

Malnutrition in South America

0

1

2

3

4

5

6

0.5 1 1.5 2 2.5 3 3.5 4 4.5 MorePercent of Severely Underweight Children

Fre

qu

ency

Figure 4: Histogram for childhood malnutrition in Latin America

highly skewed, then the median usually gives a better measure of center. However, the mean doeshave an important property not possessed by the median. As noted previously in our basketballteam example, the mean “levels off” the data set. In other words, the mean gives the value of eachdata point assuming they were all exactly the same value. For example, suppose a company has10 employees with a mean salary of $31,000 and a median salary of $27,000. The total amount ofmoney paid out in salaries is 10×$31, 000 = $310, 000. You cannot deduce this type of informationusing the median.

Measuring Spread

The mean and median give measures of center. They answer the question, “what is the mostrepresentative element of this data set.” However, two data sets with the same mean or mediancan be very different, depending on how the data is distributed around the center, or its spread.


For example, suppose we have two companies, A and B, each with five employees. The salariesfor A’s employees in thousands are

20, 20, 30, 40, 40

and for B’s employees

25, 28, 30, 32, 35.

Now, for both companies, the mean and median salaries are $30 thousand. But the salaries forcompany A have much more spread.

One measure of spread is the range, which is defined as the difference between the highes andlowest elements in the data set. The range for company A is 40− 20 = 20, and range for companyB is 35− 25 = 10.

Standard Deviation

One problem with the range is that two data sets with the same range can still be distributeddifferently. Suppose we have company C with salaries

20, 29, 30, 31, 40.

Company C has the same mean, median, and range as company A. However, more of thesalaries are closer to the mean. In some sense, there is less spread “on the average” among thisdata set.

Because of the shortcomings of the range as a measure of spread, we introduce the standarddeviation. We will typically compute standard deviations of data sets using technology. However,it is instructive to see how the standard deviation is calculated “by hand.” Here’s how it works.

Calculating the Standard Deviation of a Data Set {xi}.

1. Find the mean, x, of the data set.2. For each data point, xi, in the data set, find the difference between it and the

mean, xi − x. These are the distances from each individual data point to themean.

3. Square the differences from step 2, (xi − x)2. [Note: Since the differences aresquared, there is no distinction between data points below the mean and datapoints above the mean.]

4. Find the sum of the squared differences, Σ(xi − x)2.5. Divide the sum of the squared differences by the number of data points in the set,

Σ(xi − x)2n . (This number is called the variance of the data set).

6. Find the square root of the variance,

√Σ(xi − x)2

n . This is the standard devia-tion.34

The lower case Greek letter sigma, σ, is often used to represent standard deviation. So

σ =

√Σ(xi − x)2

n.

Let us compute the standard deviation for two of the companies mentioned previously.34The formulas that we will use for variance and standard deviation have an n in the denominator. This gives the

standard deviation for a population. There are other formulas for sample variance and standard deviation that havea n− 1 in the denominator and give the standard deviation for a sample.

50

Example 4. Compute the standard deviation of the test scores for Company A,

20, 20, 30, 40, 40

and Company C

20, 29, 30, 31, 40.

Solution: The mean, x, for both sets of scores is 30. We found the sum of the squared differencesin the right most entry of the last row in the following table.

Company A

xi xi − x (xi − x)2

20 20− 30 = −10 (−10)2 = 10020 20− 30 = −10 (−10)2 = 10030 30− 30 = 0 (0)2 = 040 40− 30 = 10 (10)2 = 10040 40− 30 = 10 (10)2 = 100

Σ(xi − x)2 400

So, σ =√

Σ(xi−x)2

n =√

4005 ≈ 8.94 for Company A. We can think of 8.94 as being roughly the

average distance of the salaries from the mean. Now for Company C:

Company C

xi xi − x (xi − x)2

20 20− 30 = −10 (−10)2 = 10029 29− 30 = −1 (−1)2 = 130 30− 30 = 0 (0)2 = 031 31− 30 = 1 (1)2 = 140 40− 30 = 10 (10)2 = 100

Σ(xi − x)2 202

So, σ =√

Σ(xi−x)2

n =√

2025 ≈ 6.36 for Company C. This means that the average distance of

the scores from the mean, 30, is roughly 6.36 or $6,360. Since the standard deviation for CompanyA is greater than the standard deviation for Company C, we say that Company A has a largerspread in its salaries.

Percentiles and Quartiles

Another measure of the spread of a data set uses percentiles or quartiles. The nth percentileof a distribution is the number such that n percent of the observations fall at or below it. Forexample, in 2005, the 20th percentile for annual houshold income in the U.S. was $18,500. Thismeans that 20% of all households earned $18,500 or less, and thus 80% of all households earnedmore than this. Here is a table of percentiles for U.S. income distribution in 2005.35

Percentile 20th 40th 60th 80thIncome $18,500 $34,738 $55,331 $88,030

35http en.wikipedia.orgwikiHouseholdincomeintheUnitedStates


From this table we get an idea of the distribution of incomes in the U.S., we cannot see therange, since the maximum and minimum incomes are not shown. We can infer that 20% of allhouseholds earn between $18,500 and $34,738, and that 60% of the population makes between$18,500 and $88,030. As a side note, percentiles that are multiples of 20 like those in this table arealso called quintiles since they divide the data into 5 equal or nearly equal groups.

By definition, the median is always the 50th percentile. Other commonly reported percentilesare the 25th and the 75th. These percentiles are also called quartiles. The 25th percentile isknown as the first quartile, the median is the second quartile, and the 75th percentile is knownas the third quartile. The 1st, 2nd, and 3rd quartiles divide a distribution into four groups withroughly the same number of data points in each group.

Percentiles, including quartiles, can often be found via technology. When finding quartiles “byhand,” the 1st quartile is defined as the median of the values below the median while the 3rdquartile is defined as the median of the values above the median. So, for example, in the data set

{2, 3, 7, 7, 8, 10, 12, 17, 18, 35}the median will be 9. For the five numbers below the median, the median is 7, and so 7 is the firstquartile. The third quartile is 17.

The difference between the 3rd quartile and the 1st quartile is called the interquartile range.This is yet another measure of the spread of the data.

Previously, we had referred to data points which are far above or far below the bulk of the dataas outliers. To be more precise, outliers are often defined as data points which are more than 1.5times the interquartile range above the 3rd quartile or below the 1st quartile.

For example, the interquartile range in data set above is the difference between 17 and 7, or10, and 1.5 times 10 is 15. The 3rd quartile is 17 and adding 15 to 17 gives 32. Thus, the datapoint 35 is an outlier since it is greater than 32.

Measuring Variability

Previously, we looked at data collection techniques and the problems of bias and variability indata collection. In this section, we will consider how to quantify variability.

Even when measures are taken to reduce bias to a minimum, any sampling technique includesthe presence of variability, unless one gathers data from every single individual in the populationunder consideration. Variability can be measured using the margin of error, which is based onthe idea of standard deviation discussed earlier.

In March of 2007, the McGovern Center for Leadership and Public Service and the DakotaWesleyan University Mathematics Department conducted a statewide poll of 410 South Dakotacitizens. One result was that 77% of those polled indicated that global warming was a serious orsomewhat serious problem.36 The margin of error for this poll was reported as plus or minus 5%.

What this margin of error indicates is that we can be 95% sure that the 77% poll result obtainedfrom the sample of 410 citizens is less than 5% away from the actual percentage of all South DakotaCitizens who would indicate that global warming is serious or somewhat serious. In other words, wecan be 95% sure that the population parameter this poll was attempting to estimate was between77 − 5 = 72% and 77 + 5 = 82%. This range, from 72% to 82% is called the (95%) confidenceinterval for this result.

One can compute margins of error and confidence intervals for all kinds of different samplingsituations, giving different types of confidence intervals. All these formulas are based on the defi-nition of standard deviation we looked at earlier. The formula used to determine a 95% confidenceinterval for a population proportion like the one above is

p̂± 2

√p̂(1− p̂)

n.

36http www.dwu.edu /press /2007 /jun13.htm

52

In this formula, p̂ is the sample proportion and n is the sample size.37

This formula actually determines two numbers, namely p̂− 2√

p̂(1−p̂)n and p̂ + 2

√p̂(1−p̂)

n . Thesetwo numbers determine a confidence interval centered at p̂ by giving the lower and upper end-

points of the interval. The number 2√

p̂(1−p̂)n is the value of the margin of error. There is a 95%

probability that the interval we obtain in this way will contain the population proportion.For Dakota Wesleyan’s global warming poll, the margin of error was calculated as

2

√.77(1− .77)

410= 2

√.1771410

≈ 0.042

or 4.2%. Adding and subtracting this from 77% we get a confidence interval of 72.8% to 81.2%.Why did DWU report a 5% margin of error? As is typical for most polls, this poll considered a

number of different questions and reported different percentages for each of these. For percentagesdifferent than 77%, the margin of error would be different than 0.042 or 4.2%. The worst possiblemargin of error for a poll is when the sample proportion p̂ = .50. In this case, the margin of erroris

2

√.5(1− .5)

410= 2

√.25410

≈ 2(0.0247) = 0.049

which is about 5%. So, all other margins of error for a poll like this with a sample size of n = 410 willbe smaller than 5%. Rather than provide separate margins of error for each sample proportion,pollsters typically provide the “worst case” margin of error, knowing that the actual individualmargins of error will be no worse than this.

Example 5. The McGovern Center poll mentioned above included 72 individuals from Min-nehaha county, which is where Sioux Falls is located, South Dakota’s largest city. We could considerthis a random sample of the population of Minnehaha county. Of these 72, 56 considered globalwarming a serious or somewhat serious problem. Find the sample proportion and the margin oferror for this poll. Do Minnehaha county residents consider global warming a more serious problemthan South Dakota citizens as a whole?

Solution: We know that p̂ = 56/72 = 0.78 and n = 72. Substituting these values into ourformula, we get a margin of error of

2

√0.78(1− 0.78)

72≈ .0976

or about 10%.The 78% sample proportion is only 1 percent higher than the 77% for all South Dakotans.

Given the 10% margin of error for Minnehaha county and the 5% margin of error for the state, wewould say there is essentially no difference between these two poll results. After all, remember thatthere is a 95% chance that the actual result will be within the margin of error. Since we are just aslikely to overestimate the actual result as underestimate, this means, for example, there is a 47.5%chance that the Minnehaha county result will be between 68% and 78%. Thus, the Minnehahacounty result could easily be below 77%, not even taking into account that the actual state resulthas a good chance of being higher than 77%. Thus, there will be a good chance, close to half infact, that Minnehaha county residents are actually slightly less concerned about global warmingthan other South Dakotans.

37Most statistics books use the more precise value of 1.96, rather than 2, in this formula. However, the errorintroduced by rounding off to 2 is not important in most cases.


Weighted Averages

Welcome to Lake Wobegon, where all the women are strong, all the men are good-looking, and all the children are above average— Garrison Kiellor

A common concern among college students, particularly as they see graduation approaching,is their GPA or Grade Point Average. Students perceive that this one number has the potentialto greatly affect their futures. It can determine whether or not a student gets into medical school,law school, or graduate school. It may influence future prospective employers who are holding thekeys to that “dream job.” For many student-athletes, their GPA is a big factor in whether theyplay or not. It is used at most schools as the benchmark by which those who graduate with honorsare separated from those who don’t.

The GPA is what is known as a weighted average. A weighted average is similar to theordinary mean, except some of the data points being averaged count more (have more weight)than others. In the case of the GPA, some classes count more than others, depending on how manycredits those classes carry.

You may already be concerned enough about your GPA to have computed it yourself. Here isan example. Suppose a student has courses and letter grades as shown in Table 1.

Table 1: Grade Information for a Hypothetical Student

Course Credit Letter Grade QualityTitle Hours Grade Points Points

General Biology 4 C 2.0 8.0Foundations of Ed 2 B+ 3.3 6.6

U.S. History 3 A- 3.7 11.1Statistics I 3 B 3.0 9.0

Tennis & Golf 1 A 4.0 4.0Human Development 3 C- 1.7 5.1

TOTAL 16 43.8

The grade points corresponding to the letter grades are given in the fourth column. Therightmost column shows what are called Quality Points in Registrar’s lingo. These are the productsof the course credits and the grade points for each class. The GPA is then calculated by adding upthe Quality Points for each class to get the total quality points, and then dividing by the numberof credits. For this student, the GPA is 43.8/16 = 2.738. (Registrars typically round the GPA tothree decimal places).

What if all the courses had been weighted the same? In other words, what if all the courseswere the same number of credits, or if we just ignored the credit hours? In this case, we would justadd up the grade points for each class and divide by 6. The result would be

2.0 + 3.3 + 3.7 + 3.0 + 4.0 + 1.76

= 2.95

Why is this higher than the actual GPA? Looking at the grades, we see that the student’s bestgrade was in a 1 credit course, and their worst two grades were in 3 and 4 credit courses. Whencomputing the GPA, these lower grades were weighted 3 and 4 times as much as the A in Tennis &Golf. When we ignore the weighting, the A counts as much as the C or the C-, and so the averagegets raised.

Another way to calculate the GPA is as follows.

416· 2.0 +

216· 3.3 +

316· 3.7 +

316· 3.0 +

116· 4.0 +

316· 1.7 = 2.738

54

Here, we are dividing the credit hours for each course by the 16 total, instead of doing themultiplying and adding first, and dividing at the end. Each grade is “weighted” by the fractionof the 16 total credit hours represented by that class. You could even convert these fractions topercentages. In fact, many instructors use this type of weighted average approach in calculatinggrades for their courses. Each exam, quiz, paper, etc. is given a percentage weight, and this ismultiplied by the grade each student earns on that item, and then all the items are added up.

There are many other examples of weighted averages. Although the methods for calculatingthese might be modified from the basic example above and be more complicated as a result, theyall have in common the idea of weighting. Oftentimes, numbers that are the result of a weightedaverage type of calculation are called indexes (singular index, and sometimes the plural is indices).Examples include:

1. Stock indices, like the Dow Jones Industrial Average, the NASDAQ Composite Index, andthe S&P 500 Index.

2. Football quarterback ratings.

3. The Consumer Price Index (CPI) of inflation.

4. Many, many other economic indexes, like the Consumer Confidence Index, and the Index ofLeading Economic Indicators.

5. The Social Capital Index, introduced in Robert Putnam’s Bowling Alone.

6. Other social indexes, like the United Nations Human Development Index.

7. Scoring at figure skating competitions

Some of these indexes are “just for fun.” However, economic and social indexes are veryimportant, because they are used by governments, companies, and even individual citizens inmaking decisions. For instance, many companies and organizations (including DWU) base salaryraises on the CPI. Investment decisions usually take stock indexes into account. Economic indexescan decide who wins and who loses in presidential elections.

As is the case with the usual arithmetic mean, indexes are designed to give a single numericalmeasure of a particular variable, based on a larger (sometimes quite large) set of data. For example,the S&P 500 gets part of its name from the fact that it uses data from 500 companies in computingthe index. The Consumer Confidence Index is based on responses to five questions on a surveycollected from thousands of individuals.

The following examples will give you an idea of how indexes are designed and calculated. Asyou read through these, think about why the these indexes are designed the way they are. Howmight they have been designed differently? Does it seem reasonable that the index is actuallymeasuring what it is intended to measure? In the activities for this chapter, you may have theopportunity to design your own index, and you can use these examples as models to follow.

Example 6. One of the most famous averages in existence, The Dow Jones Industrial Average(DJIA), was introduced on May 26, 1896 by Charles H. Dow. At the time,“stocks moved on dubioustips and scurrilous gossip because solid information was hard to come by.” Originally, the DJIAwas calculated as the average price of 12 stocks hand-picked by Dow. Dow hoped his average wouldbe a good overall measure of how the stock market was performing. 38

The DJIA has become a bit more complicated over the years. Today, the editors of TheWall Street Journal (WSJ) select a list of 30 large U.S. companies to calculate the DJIA. Thesecompanies would be called the components of the index. The stocks prices of these companies areaveraged, but the average is adjusted to take into account changes that have occurred over time.For example the average is adjusted for “splits,” which occur when a company decides to issue twoshares of new stock for each old share, with the price of the new shares half of the old price. If one

38information on the DJIA, NASDAQ, and S&P 500 indices can be found at www.investopedia.com/ andwww.cbot.com


of the 30 stocks is split, then that would lower the price of that stock, and thus lower the average.Also, there have been additions to and substitutions in which stocks have been used to computethe average over the years. For example, when one of the companies goes out of business, or ispurchased by another company, this company would be replaced by another as a component. Infact, General Electric is the only company remaining from the original list. When this happens,the stock price for the substituted company is probably not the same as the company that wasdropped from the index.

To compensate for these types of changes, the WSJ uses a “Dow Divisor”. The sum of the30 stock prices is divided by this Dow Divisor, rather than the number of stocks (30). If a stocksplits, resulting in a lower average, then divisor is also lowered in such a way so that the DJIAdoes not change as a result of the split. The divisor is also adjusted when changes are made in thecompanies that are selected, in order that these changes by themselves do not produce any changein the actual DJIA index. As of March 1, 2004, the divisor was 0.13500289. The changes thathave occurred over the years have substantially decreased the divisor! By adjusting the divisor aschanges occur, it is hoped that the DJIA is a good measure of the stock market over time.

Example 7. Since the individual stock prices are not weighted, the DJIA is not really a weightedaverage. However, most other stock indexes are, including the S&P 500 and the NASDAQ compos-ite. In these indexes, the stock prices are weighted by what is known as (market)-capitalization.The market-capitalization of a company is the total value of all its stock, and is one measure ofwhat a company is worth, overall. It can be computed by taking the current stock price times thecurrent number of shares. For example, a company with 2 million shares and a stock price of $50per share has a capitalization of $100 million. Another company with 10 million shares and a stockprice of $25 has a capitalization of $250 million.

An index that is weighted by market-capitalization would take the stock price times the capi-talization for each company that is a component of the index, add these together, and then divideby the total market capitalization of all the companies. Thus, companies that have a higher over-all total worth are weighted more in the index since there stock price gets multiplied by a largernumber. For example, suppose we have only two companies represented in our index, company Awith a capitalization of $5 million and a stock price of $20, and company B with a capitalizationof $15 million and a stock price of $10. Our index value would the be

$5 · 20 + $15 · 105 + 15

=25020

= 12.50

Notice this is smaller than the average of the share prices, which would be (20 + 10)/2 = 15.This is because the company with the lower stock price has the larger capitalization, and so isweighed more in the index, making the index lower. If company B loses a lot of capitalization andits total capitalization goes down to $5 million (same as company A), but it still has the samestock price of $10, the index would be

$5 · 20 + $5 · 105 + 5

=15010

= 15.00

the same as the average of the two prices.A capitalization weighted index is considered a much more accurate reflection of the overall

stock market than the DJIA. However, because of tradition, the DJIA is still the most prevalent.As a final caution, consider the following quote. Although indexes can be very useful, we should

usually be wary of placing too much credence on any single number, no matter how well it seemsto reflect what we want to measure.

There are difficulties with all economic statistics, but the problems do not arise becausethe people who designed them were stupid or lazy. The problems arise because we aretrying to use a single number to summarize a phenomenon more complex than anysingle number can report.— Robert Schenk, Professor of Economics, St. Joseph’s College, Indiana

56

Example 8. As a less serious example of an index, let’s consider the ubiquitous quarterbackrating system that is used for both NFL and college football quarterbacks. We will refer to it asa rating instead of an index, although the idea is the same. The NFL would remind us that thisrating is really only designed to measure the passing proficiency of the quarterback, not his overallperformance. Here’s how it works.

The rating has four components.

1. Percentage of completions per attempt C

2. Average yards gained per attempt Y

3. Percentage of touchdown passes per attempt T

4. Percentage of interceptions per attempt I

The rating formula is designed so that high values for C, Y , and T increase the rating, whilehigh values of I (interceptions are bad) lower the rating. With a couple of caveats,39 the rating iscomputed using the following formula.

Rating =1006

(0.05(C − 30) + 0.25(Y − 3) + 0.2T + (2.375− 0.2I))

The caveats are that if any of the four terms 0.05(C − 30), 0.25(Y − 3), 0.2T , or (2.375− 0.2I)are bigger than 2.375, or less than 0, then substitute the numbers 2.375 or 0 respectively.

As an example, suppose that a quarterback has a completion percentage of C = 65, an averageyards per attempt of Y = 13.2, a percentage of touchdowns per attempt of T = 8 and a percentageof interceptions of I = 2. Putting these into the formula, we get

1006

(0.05(65− 30) + 0.25(13.2− 3) + 0.2 · 8 + (2.375− 0.2 · 2))

=1006

(1.75 + 2.55 + 1.6 + 1.975)

Since the 2.55 is bigger than 2.375, we replace it with 2.375 and get

Rating =1006

(1.75 + 2.375 + 1.6 + 1.975) = 128.3

Our hypothetical quarterback is very, very good. In practice, it is very hard to get one ofthe terms to be larger than 2.375, at least over a whole season, although it does happen whenconsidering only single games. In our case, we put in 13.2 yards per attempt to get the 2.55 term.The best quarterbacks in the league typically do not average even 10 yards per attempt over aseason. In 2003, the best season long rating was 100.4 by Steve McNair. The highest possiblerating is 158.

Let’s look again at the formula.

Rating =1006

(0.05(C − 30) + 0.25(Y − 3) + 0.2T + (2.375− 0.2I))

The weighting is achieved chiefly by the numbers 0.05, 0.25, 0.2, and 0.2. In fact, these weightsand the subtractions that occur are actually designed to give roughly equal weight to each of thefour components. To achieve the top rating, a quarterback must make all the component termsequal to 2.375.

Notice the I is multiplied by a negative number. This reflects that a quarterback wants thisnumber to be as low as possible, rather than as high as possible. The 100 and the 6 are presentmerely because the NFL wanted a rating of 100 to represent a very good quarterback (one whois “100%”). The NFL could have left these numbers out, and the result would have been thatexcellent passers would have ratings of 6 or more instead of around 100 or more.

39or exceptions


Averages of Averages

Before we leave behind discussion of averages, there is one more item that is important todiscuss. What happens if the data that you are averaging are themselves averages?

Suppose you own a small business that operates in two locations, A and B. Suppose the averagecompensation that you provide for employees at A is $20,000, and at B it is $30,000. The average ofthese two numbers is $25,000. Does this mean that the average compensation for all your employeesis $25,000?

Not necessarily! There is an important piece of information that is missing, and that is howmany employees work at each location. If both locations had the same number of employees, thanthe answer is yes. Suppose, however, that there are 8 employees at location A, and only 2 atlocation B. Then the total amount of compensation received by all 10 employees is

8($20, 000) + 2($30, 000) = $220, 000

and so the average is $220, 000/10 = $22, 000. It is lower than the $25,000 since there are moreemployees at the location that has the lower average salary. If the situation were reversed andthere were 8 employees at B and only 2 at A, then the average for all employees would be $28,000.

The moral of the story is that you need to beware when you are “averaging averages.” It maysound silly, but remember that

the average of the averages is not necessarily the average

This idea will be relevant in many of the situations we consider. For example, we have alreadypresented some data in graphical form based on the 50 states. If our data is in the form of averages,say average household income in each state, then the average of these 50 numbers is not the sameas the average household income of all households in the U.S.

If a large state (like California, which has about one fifth of all the households in the U.S.) hasa lower income than most states, than it will cause the overall U.S. average to be lower than theaverage of the 50 state averages, in the same way that the employees at location A (which hadfour fifths of the employees) caused the overall average for the 10 employees to be lower than theaverage of the two location averages. The only difference is we now have 50 averages that we areaveraging, instead of just two.

In fact, the same idea of “weighting” that we introduced with the GPA example applies here.When averaging the 50 state income averages, the California data point is weighted equally withall other 49 states. However, if we actually count the households in each state, than the Californiaincome number would be weighted more than all the other states, about one fifth of the totalweight. A small state like South Dakota would have a weight of about 1/375, based on the factthat about 1 out every 375 households in the U.S. is in South Dakota.40

40So, unfortunately, in some sense we don’t count for much. On the other hand, there are advantages, since wedon’t see nearly the number of negative political ads as they do in California

Reading Questions for Collecting and Analyzing Data

1. What does it mean for a sample to be a simple random sample?

2. What does it mean for a sampling technique to be biased?

3. A very conservative talk radio host decides to conduct a survey in his city. He hires a localsurvey firm to conduct a random sample of 500 residents of the city. One of the survey resultsis that 48% of those surveyed support a proposed constitutional amendment outlawing gaymarriage. Is this sample biased? Why are why not?

4. A survey firm conducts surveys on customer satisfaction with local cable service every twoweeks for an entire year. Assume that the actual level of customer satisfaction is approxi-mately constant 64% over the whole year. The 26 survey results were as follows:

52, 66, 63, 75, 59, 84, 68, 56, 49, 67, 60, 72, 71, 66, 54, 61, 58, 63, 79, 64, 68, 45, 58, 73, 66, 70

(a) Does this situation show high bias, low bias, or is it impossible to tell? Explain.(b) Does this situation show high variability, low variability, or is it impossible to tell?

Explain.(c) Does this survey firm seem to be producing reliable data? Why or why not?

5. An elementary teacher has her class measure the height of each student in “hands.” Shepairs off the students. Each student measures the other student in their pair using their handwidth, counting the number of hand width they need to go from foot to head.

(a) If this measuring technique is used repeatedly, mixing up the pairs each time, would thetechnique produce reliable data? Why or why not?

(b) Does this technique produce valid data? Why or why not?(c) Suppose that, instead of pairing students up, the teacher picks one student to measure

all the other students. Would this technique produce valid and reliable data?

6. If you repeat a sample several times and get very similar answers, you have (low, high, can’ttell) variability and (low, high, can’t tell) bias.

7. A local environmental organization likes to include polls in its fund-raising mailings becauseit seems to increase response rates and the amount raised. Often, they ask pretty much thesame questions from one poll to the next. The polls get sent to everyone on the organizationsmailing list. One poll question that has been asked several times is whether congress shouldpass laws to mandate a reduction in greenhouse gas emissions. The results from the polls areconsistently that 75% to 80% of the membership agree with this statement.

(a) Does this situation shows high bias or low bias? Why?(b) Does this situation show high variability or low variability? Why?(c) Does the sampling technique used in this situation produce simple random samples?

8. Identify the population and the sample in each of the following situations.

(a) A biologist wants to know the mercury level present in a certain species of fish in LakeCatalano. Over the course of two days, she traps 37 fish live, collects a blood and tissuesample from each, and returns the fish to the lake. She finds that 17% of the fish sampledhave mercury levels higher than recommended by the EPA.

58


(b) On July 31st, 2006, CNN conducted a web poll asking if people felt the term ’‘tarbaby,’ used by Massachusetts Governor Mitt Romney to describe Boston’s ‘Big Dig’construction project, was a racist term. Of 63,240 votes registered as of 3:30 p.m. CDT,45% responded ‘yes.’

9. Find the median for each of the following.

(a) 76, 52, 46, 55, 49, 80

(b) 78, 83, 96, 58, 95, 87, 91

(c) What did you do differently for part (b) as compared to part (a)?

10. A fund-raiser for an elementary school organization brings in the following amounts of money:

$60, $125, $80, $25, $130, $90, $50, $75, $40, $140

All the money is to be used for a trip and the adult leader decides that each of the 10 childrenwill get the same amount of money. How much does each child receive?

11. List 8 numbers that have exactly one outlier.

12. List 6 numbers whose mean is smaller than the median.

13. List 10 numbers whose histogram will be skewed to the right. Sketch the histogram for yourdata.

14. Suppose there are nine people working in a dentist’s office. The three people that keep trackof the patients and the books each earn $36,000 per year, the four dental hygienists each earn$42,000 per year, and the two dentists each earn $106,000 per year. What is the mean andthe median salary of people working in this office? Which is the better measure of center inthis case? Explain.

15. When is it better to use the median instead of the mean as a measure of center?

16. Points X, Y , and Z are labelled on the following histograms. Determine if the labellednumber represents the mean, median, both, or neither.

(a)

2

4

6

8

10

12

14

X0 10 20 30 40 50 60 70

Collection 1 Histogram

60

(b)

2

4

6

8

10

Y0 10 20 30 40 50 60


(c)

2

4

6

8

10

Z20 30 40 50 60 70 80 90


17. What does it mean to be at the 80th percentile for ACT scores?

18. Find the standard deviation and the interquartile range for the following sets of data (youmay use a calculator or software):

(a) 20, 20, 21, 23, 26, 27, 27, 30, 31, 31, 32

(b) 10.1 10.8 12.6 13.1 14.3 14.4 14.4 14.6 14.7 15.1 16.0 16.3 16.8 18.1

19. Consider the histogram on University of Maryland faculty salaries given in Figure 8 of ChapterTwo.

(a) Looking at the histogram, estimate as accurately as you can the minimum, first quartile,median, third quartile, and maximum.

(b) Looking at the histogram, and considering its skew, estimate the mean salary for Uni-versity of Maryland faculty.


20. Consider the histogram on National Poverty Rates given in Figure 10 of Chapter Two.

(a) Looking at the histogram, estimate as accurately as you can the minimum, first quartile,median, third quartile, and maximum.

(b) Looking at the histogram, and considering its skew, estimate the mean national povertyrate.

21. A student has scored 78, 89, 86, and 72 on the first 4 tests in her accounting class. There willbe one more test, and all tests are out of 100 points. Her grade will be based on the averageof all 5 tests.

(a) What is the highest average this student can possibly achieve?(b) What is the lowest average this student can possibly achieve?(c) What score does she need to get on the last test to get an average of 80?

22. From Figure 10 in Chapter Two, carefully estimate both the mean and the median NationalPoverty Rate for the countries represented.

23. The following table shows childhood mortality data for a number of Eastern European coun-tries for the year 2001. The data is the number of deaths of children five and under per 1000live births. 41 Use this to answer the following questions.

Deaths per Deaths perCountry 1000 births Country 1000 births

Czech Rep. 5 Bosnia and Herzegovina 18Germany 5 Yugoslavia 19Slovenia 5 Belarus 20Croatia 8 Ukraine 20Hungary 9 Latvia 21Lithuania 9 Romania 21Poland 9 Russian Federation 21Slovakia 9 TFYR Macedonia 26Estonia 12 Albania 30Bulgaria 16 Moldova, Rep. of 32

(a) Without doing any calculations, which of the following is most likely to be true? Brieflyjustify your answer.• The mean will be greater than the median.• The mean will be less than the median.• The mean will be almost equal to the median.

(b) Compute the mean and the median for this data.(c) Compute the standard deviation for this data.(d) Draw a histogram for this data, and then mark the mean and median along the horizontal

axis.(e) Compare your histogram with the mortality histogram in Figure 8 of Chapter 2. How

do mortality rates in Eastern Europe differ from mortality rates for the world overall?

41Data is from the UNICEF end of decade database at http://www.childinfo.org/eddb/malnutrition/index.htm.

62

24. As noted in the text, one must be careful to interpret what an average means when thenumbers being averaged are based on different sized populations.

(a) Find the average of the mortality rates for the four countries Estonia, Latvia, Lithuania,and Russia.

(b) Now, use the table below, which gives the total number of births (in thousands) in eachof these countries, and the information in the previous question to find the total numberof child deaths in each country.

Live BirthsCountry (in thousands)Estonia 30

Lithuania 12Latvia 18Russia 1230

(c) Now, find the total number of deaths in the four countries combined and the totalnumber of births (in thousands) combined. Use this to find the overall child mortalityrate in these four countries.

(d) Why is the overall rate from part (c) so much bigger than the average of the four ratesyou found in part (a).

25. Using your calculator or software, find the median, quartiles, mean, and standard deviationof the following frequency distribution of student heights. Write a brief sentence describingeach of the results.

Height 66 67 68 69 70 71 72Frequency 1 3 5 8 7 3 2

26. Suppose we added two individuals to the height data in the previous question, one who is67 inches tall, and one who is 81 inches tall. How will these additions change the meanand median of the data? Explain how you can answer this question without necessarilyrecalculating both values.

27. Consider the quote from Rush Limbaugh a the beginning of this chapter. This quote occurredwithin a section of his show where he was arguing that middle class Americans had not sufferedduring the so-called “decade of greed”, the 1980’s, during which Republicans Ronald Reaganand George Bush (senior) were President.

(a) Explain why it is reasonable to assume that the distribution of family incomes in theU.S. is skewed to the right.

(b) Given that incomes are skewed to the right, which is larger, the median family incomeor the mean family income?

(c) Explain why it possible for the mean income to increase over time, while the medianincome remains the same.42

42In fact, according to The Statistical Abstract of the United States, from 1982 to 1989 real median income for allU.S. households went up 10.4%. Mean Per Capita Income went up about 20%. This number is a mean, since it isper capita, and the denominator is the total population. The median computation is done per household. By way ofcomparison, households at the 95th percentile (those earning more than 95% of all households) had their income goup by 17%, while those at the 20th percentile went up 8%, over the same period.


28. According to Bread for the World, a hunger advocacy organization, the average U.S. farmfamily earned $64,117 in 2002, almost $6000 more than the average American family. How-ever, the same publication also notes that half of all farmers earn less than $6000 per year infarm income, and many farmers make most of their family income from non-farm work.43

(a) Do you think the $64,117 figure is a mean or a median? Explain why you think so.(b) Suppose that, based on the information above, someone made the claim that “The

average American farmer earns $6000 per year from their farming operation and $58,117in non-farm related income.” Explain carefully why this statement is probably notcorrect.

29. In the poll mentioned in Example 4 in the text, there were 20 respondents from the 7 countyarea around Mitchell, South Dakota. Of these, 16 indicated they thought global warmingwas a serious or somewhat serious problem.

(a) Give a 95% confidence interval for the proportion of all the people in the Mitchell areathat think global warming is a problem.

(b) State carefully what this confidence interval means in terms of probabilities.

30. Suppose a student scores 82% on quizzes, 76% on tests, 89% on his project, and 78% onhis final. Suppose these are weighted 25%, 40%, 15% and 20% respectively. His final gradepercentage is the weighted average. Find the student’s final grade percentage.

43Statements are from Bread for the World’s 2003 annual report, Agriculture in the Global Economy

64

Collecting and Analyzing Data: Activities and Class Exer-cises

1. The Curse Reversers. Below is a table showing the salary for members of the 2004 RedSox, the only team to come back from a 3 games to none deficit in a championship series inbaseball history.44 They are also given credit for reversing the ‘curse of the Bambino,’ placedon the Red Sox after then owner Harry Frazee sold Babe Ruth to the New York Yankees in1920 for $100,000. The 2004 team beat the curse by being the first Red Sox team since 1918to win the World Series.45

Player Salary Player SalaryRamirez, Manny 22,500,000 Embree, Alan 3,000,000Martinez, Pedro 17,500,000 Timlin, Mike 2,500,000Schilling, Curt 12,000,000 Mueller, Bill 2,100,000Garciaparra, Nomar 11,500,000 Reese, Pokey 1,000,000Damon, Johnny 8,000,000 Mirabelli, Doug 825,000Varitek, Jason 6,900,000 Burks, Ellis 750,000Ortiz, David 4,587,500 Kapler, Gabe 750,000Lowe, Derek 4,500,000 Daubach, Brian 500,000Nixon, Trot 4,500,000 McCarty, David 500,000Wakefield, Tim 4,350,000 Bellhorn, Mark 490,000Mendoza, Ramiro 3,600,000 Arroyo, Bronson 332,500Foulke, Keith 3,500,000 Crespo, Cesar 309,000Kim, Byung-hyun 3,425,000 Shiell , Jason 303,000Millar, Kevin 3,300,000 Garcia, Reynaldo 301,500Williamson, Scott 3,175,000 DiNardo, Lenny 300,000

(a) Without doing any calculations, decide if the median salary or the mean salary will behigher. Explain how you determined your answer.

(b) Calculate the mean, median, and standard deviation of the salaries for the 2004 RedSox using a calculator or software.

(c) Assume players on the Red Sox are paid according to their abilities. What should aplayer of average ability expect to get paid? Explain your answer.

(d) What percent of Red Sox players made below the mean salary? What percent madebelow the median salary?

(e) Suppose each player on the Red Sox was given a $50,000 pay raise. Without recalcu-lating, determine the new mean, median, and standard deviation. Explain how youdetermined your answers.

(f) What percent of the payroll was earned by Manny Ramirez? Starting at the bottom ofthe pay scale, how many players does it take to get a total salary equal or higher thanManny’s?

44The salary data is from <http://asp.usatoday.com/sports/baseball/salaries/playerdetail.aspx?player=1684.>45For more information on ‘the curse’, see <http://www.bambinoscurse.com/whatis/.>


2. Hungry Planet II. The table below gives some of the data for the 30 families surveyed inHungry Planet. For each family, the table gives the country, number of adults, children, andtotal persons, the total weekly food costs for the family, and the cost per person.

Country Adults Children Persons Weekly Cost Per PersonAustralia 3 4 7 376.45 53.78Australia 2 2 4 303.75 75.94Bhutan 6 7 13 34.09 2.62Bosnia 2 3 5 157.43 31.49Chad 1 5 6 25.59 4.27Chad 2 7 9 43.77 4.86China 3 1 4 155.06 38.77China 5 1 6 59.23 9.87Cuba 2 2 4 69.92 17.48Ecuador 2 7 9 34.75 3.86Egypt 5 6 11 68.53 6.23France 3 1 4 419.95 104.99Germany 2 2 4 500.07 125.02Great Britain 2 2 4 253.15 63.29Greenland 2 3 5 498.1 99.62Guatemala 2 6 8 79.82 9.98India 2 2 4 39.27 9.82Italy 2 3 5 260.11 52.02Japan 2 2 4 317.25 79.31Okinawa 3 0 3 221.51 73.84Kuwait 4 4 8 221.45 27.68Mali 4 11 15 26.39 1.76Mexico 2 3 5 189.09 37.82Mongolia 2 2 4 40.02 10.01Philippines 5 3 8 49.42 6.18Poland 4 1 5 151.27 30.25Turkey 3 3 6 145.88 24.31United States 2 2 4 159.18 39.80United States 2 2 4 341.98 85.50United States 3 2 5 242.48 48.50

(a) Find the five number summary for the weekly per person costs. Write a few sentencesdescribing what these data tell you about what it costs to feed a person for a week.

(b) Note that the average of the per person costs represents an average of the family averages.By summing up the number of persons in each family, we get that there are 183 peoplein these 30 families. What is the average cost per person for these 183 people, and whyis it different than the average you found in part (a)?

(c) Find both the average of the family sizes and the average number of people per familyand explain why these are the same.

(d) Does this data support the statement that large families tend to spend more total onfood than small families? Use graphs or appropriate statistics to support your answer.

(e) Explain as thoroughly as possible why the data represented by these 30 families isprobably not representative of average food costs world wide.46

46As a hint, note that China and India are the most populous countries in the world, each with over 1 billionpeople. The United States currently has around 300 million people.

66

3. Homelessness. We noted in the text that reports that the average age of a homeless personis nine have been widespread. Adam Molnar47, a mathematics professor at Bellarmine Uni-versity, found that this claim originated with ads created by the Coalition for the Homeless,which bills itself as the oldest homeless advocacy and direct service organization in the U.S.48

(a) Create a set of five ages where the average age is nine, and then discuss how likely youthink it would be to find a group of five homeless people with these ages.

(b) Create a set of five ages where the average age of the five people is nine, and one of thepeople is 40 years old. Would you be likely to find a group of five homeless people withthese ages?

(c) Using Fathom, create a data set of 25 ages and an associated histogram so that theaverage of the 25 ages is nine. Use at least 10 different ages, and include at least 5adults in your histogram.

(d) Describe the skew and other characteristics of your histogram.

(e) How likely do you think it is that this claim is true? Describe how you might go aboutdetermining whether or not it is true.

4. City Versus Highway. The graph below represents the relationship between HighwayMileage and City Mileage for 40 different car models.

Highway Mileage Versus City Mileage

20

24

28

32

36

40

16 20 24 28 32

(a) What type of graph is this?

(b) Even without the graph heading, how can you tell that highway mileage is representedon the vertical axis?

(c) What are the highway gas mileages for cars which get 21 miles per gallon in the city?

(d) Every car gets better mileage on the highway than in the city. What are the highwayand city mileage ratings of the models that have the smallest difference between cityand highway?

(e) What are the highway and city mileage ratings of the models that have the largestdifference between city and highway?

(f) There are only 22 points shown on the graph, even though the graph represents 40different models. How is this possible?

47see http://mathematicsafterthefall.twelvefruits.com/ for Adam’s blog entry on this claim.48http://www.coalitionforthehomeless.org/home.asp


5. Gross Domestic Product. Gross Domestic Product. Below is a table showing the PerCapita Gross Domestic Product (GDP) for a number of countries for the year 2002. The GrossDomestic Product is an estimate of the total value of all the goods and services producedin a country, and is considered one measure of the total wealth of a country. Per CapitaGDP (PC GDP) is the GDP divided by the population, and is considered a measure of theproductivity of a country.

Country PC GDPUnited States 36123

United Kingdom 26376Trinidad and Tobago 7109

Venezuela 3760Uruguay 3645Turkey 2626Tunisia 2163

Yugoslavia 1459Turkmenistan 1383

Tonga 1345Ukraine 849

Zimbabwe 640Yemen 559

Vietnam 436Uzbekistan 383

Zambia 352Uganda 251

(a) Without doing any calculations, use the table to decide if the median PC GDP or themean PC GDP will be higher. Explain how you determined your answer.

(b) Now, calculate the mean, median, and standard deviation of the PC GDP. How manycountries lie within one standard deviation of the mean?

(c) Consider how you would set up a histogram for this data. Without drawing it out,explain how the histogram would look (you could create a frequency chart to help).Why is a histogram probably not a very good way to display this data?

(d) List the minimum (smallest number), 1st quartile, 2nd quartile, 3rd quartile, and max-imum (largest number) for this data.

(e) Determine which countries are outliers with respect to the PC GDP, and explain why.Without doing any calculations, discuss how the mean and median would change if thesecountries were omitted from the data.

(f) Recalculate the mean, median, and standard deviation without the countries that youidentified as outliers. How do these calculations compare with your conjectures frompart e?

(g) Based on what you have done so far, explain in a few sentences what this PC GDP dataindicates. Mention particular countries as you feel appropriate.

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6. Textbook Prices. College textbooks can be both fat and expensive. The table below givesdata on 13 different calculus texts. 49

Author Pages PriceStewart 1127 180.95Larson 1138 123.35Larson 760 119.31Stewart 1336 147.62Hughes-Hallett 528 97.35Hughes-Hallett 688 117.75Lial 864 126.67Bittinger 673 121.33Hass 922 106.67Foerster 778 100.57Berresford 672 111.49Varberg 864 102.01Goldstein 736 104.14

(a) Sketch a scatterplot of the data. How would you describe the relationship between priceand the number of pages in a calculus textbook?

(b) How do you think the relationship would be different if we picked a different subject,like literature for example?

(c) How would you expect the relationship to change if picked textbooks from several dif-ferent subjects? Explain.

(d) What is the standard deviation for the price data?(e) Based on this data, how much would you budget for a calculus text if you were planning

on taking the class next fall, not knowing which textbook would be required? Explainyour reasoning.

(f) Estimate how much a page of calculus text costs, based on these data. Be sure to explainhow you arrived at your answer.

49Data acquired from amazon.com, August 14th, 2007


7. Basketball Salaries. The table below lists the teams in the National Basketball Associationalong with their total team payrolls in millions of dollars and the winning percentage for the2004-2005 season.50 We want to determine if the team owners are getting their money’sworth from their highly paid players.

Team Payroll 04-05 Winning(in millions) Percentage

New York 102.44 0.402Dallas 91.55 0.707

Portland 83.67 0.329Philadelphia 71.95 0.524Minnesota 70.06 0.537Memphis 67.01 0.549Orlando 66.45 0.439Indiana 65.79 0.537

L.A. Lakers 65.06 0.415Boston 64.58 0.549

Sacramento 61.81 0.61Toronto 61.70 0.402Houston 60.22 0.622Miami 58.95 0.72

Chicago 57.28 0.573Milwaukee 57.14 0.366

New Orleans 56.57 0.22Golden State 54.94 0.415New Jersey 54.73 0.512

Detroit 54.57 0.659Seattle 53.82 0.634

Washington 49.55 0.549Cleveland 49.18 0.512

San Antonio 47.15 0.72Denver 45.62 0.598

L.A. Clippers 45.17 0.451Phoenix 44.26 0.756

Utah 43.16 0.317Atlanta 40.68 0.159

Charlotte 23.38 0.22

(a) Looking through the table, does it seem as though the higher paid teams had higherwinning percentages? If so, were there exceptions to this rule?

(b) Make a scatterplot of this data with the team payroll on the horizontal axis and thewinning percentage on the vertical axis.

(c) Does your scatterplot have a positive or a negative association?(d) Which team would you say is the best value for the money? Why?(e) Which team would you say is the worst value for the money? Why?(f) Make a histogram of the salary data. What skew does this histogram have? What does

this tell you about team payrolls in the NBA?

50From Patricia Bender’s Various Basketball Stuff at <http://www.dfw.net/ patricia>.

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8. Standard Deviation. Given the set of numbers {6, 7, 8, 9, 10}, you are to choose fourof them with repetitions allowed (in other words, you can choose a particular number morethan once).

(a) Give a list of four of these numbers so that the standard deviation is as small as possible.Is your list unique or is there another list of four that will have the same standarddeviation? Explain.

(b) Give a list of four of these numbers so that the standard deviation is as large as possible.Is your list unique or is there another list of four that will have the same standarddeviation? Explain.

(c) How would your answers change if the set of numbers is {106, 107, 108, 109, 110} insteadof the original set?

9. Exam Grades. A college professor states that he will curve exams 10 percentage points ifthe class average is below 70. Suppose the test scores are:

75, 51, 60, 97, 90, 30, 62, 89, 83, 78, 25, 77, 80, 70, 82, 59, 83, 85, 61, 92, 78, 82, 90, 77, 78

(a) What is the median test score?(b) What is the mean test score? Will the professor curve the exam?(c) What are the percentages for the first and third quartiles? What is the interquartile

range?(d) What is the standard deviation? What does this tell you about the spread?(e) Are there any outliers? If so, list them.(f) Do you think the professor should change his syllabus and use the median test score

instead of the mean for curving exams? Explain.

10. Creating an Index I. In this activity, you will have the opportunity to create your own“car rating index” following the ideas outlined in the text. You will use the Fathom file CarIndex Data. When you open this file you should see the collection and a short descriptionof the data at the top. Read through these notes to get an idea of the variables/attributesrepresented in the data.On the left are several “sliders” as they are called in Fathom. These can represent eitherattributes (variables) or parameters that can be incorporated into formulas, and can bedynamically changed by grabbing and dragging. Try playing around with the slider labelledA a bit, dragging the grey slider icon back and forth. Notice how the value given in the upperleft of the slider box changes.

(a) Click on the formula in the Index attribute (click in the box directly under the attributename). The Formula dialog box should pop up. Write down the formula for futurereference. Which attributes and sliders are involved in this formula?

(b) Notice that most of the attributes are multiplied by a slider value. However, in theformula, we divide by the AvgPrice attribute. This is because, everything else beingequal, we would rather pay less for a car than pay more. When we divide by the price,the higher the price, the smaller the number we get, and the less our rating goes upbecause of the price.

(c) Sort the data from highest to lowest Index value. Which model has the highest rating?Which has the lowest? Which has the median rating?

i. The highest rated model should be the Toyota Prius. Consider each of the fourcomponents that are part of the index. How much does each component contributeto the Index value? To do this, calculate each of the four summands of the formulaseparately. Which component is giving the biggest contribution towards the Prius’rating?


ii. Now, look at the lowest rated model and explain why it is rated so low, againconsidering each of the four components and the data for this model.

iii. We could change the values of Index by adjusting the sliders. Try changing thevalues of the sliders so that the model that was rated second moves ahead of thePrius, and becomes the highest rated model. Which sliders did you increase inmaking this happen, and which attributes are these sliders associated with? Whichdid you decrease?

(d) The Index formula as it was originally set up involved only four attributes. We couldcertainly get different ratings if we used more attributes, or different attributes, orchanged the values of the sliders.

i. Pick your favorite model. Then, pick the four attributes, of those that are given,that you think you can use to create an Index which will give your car the highestrating. For example, if your model is one of the least expensive models, you probablywant to include AvgPrice as a component in your formula.

ii. Create a new Index formula, using the four attributes you selected in the previousquestion as components. You can do this by clicking on the Index formula andediting the formula to include only the attributes you selected. You may also wantto adjust parts of the formula to weight certain components more or less (althoughyou can also do this with the sliders).

iii. Now adjust the sliders, trying to get your favorite model to be the highest ratedmodel. Note that it may not be possible to make your model the highest ratedmodel, so just try to get it as high as possible. Once you have done the best youcan, record the formula you used and the values of the sliders as well.

(e) Finally, discuss any problems or shortcomings with this method of rating cars, includingany deficiencies in the data set. You might consider what other attributes would havebeen good to include in the data set.

11. Creating an Index II. In this activity, you will pick your own variable to measure andcreate an index for that variable. You can pick any variable you want subject to the followingconditions.

1. Do not use any of the examples outlined in the text.2. Your index should involve at least four components.

(a) Explain what variable you are measuring, and why you picked it.(b) Say which components you are including in your index and why you decided on these.(c) Decide how you are going to measure each component. For a component to be included

in a numerical index, you need to be able to assign a numerical measure to it.(d) Decide how you want to initially weight each component. One way to do this is to use

percents as weights, and have all the weights add up to 100%. If, for example, youthink a particular component should count as much as all the rest put together, thenyou could weight it 50%.

(e) Decide if there are any other adjustments to be made to the computation of your index,other than the weights. (See the NFL quarterback rating scheme as an example).

(f) Find at least three actual data points to try out your index. For example, if your variablewas “level of democracy in a country”, you might pick Sweden, Somalia, and Senegalas three sample countries, and use information for each of your components for thesecountries.

(g) Consider how your index rated the data points from part e). Were these the results youexpected? Do you feel that the data points have been ranked appropriately? Based onthe results, would you alter how your index is computed in any way?

(h) Summarize what you feel are the advantages and disadvantages of your index.

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12. The 2008 Election. In 2008, the U.S. elected its first President of African-American back-ground.51 Going into election day, there were a number of states designated as “battlegroundstates” because polls indicated those states were too close to call with any confidence. Mostpundits included Florida, Indiana, Missouri, North Carolina, Ohio, and Virginia in this cate-gory. The table below gives poll results from shortly before the election, where (O) gives thepercentage of prospective voters supporting Obama in the poll, and (M) McCain. These pollresults are from Zogby International, have sample sizes of about 600, and can be accessed athttp://www.zogby.com/news/ReadNews.cfm?ID=1632.

Last Last Elec. Elec. Elec.State Poll (O) Poll (M) Result (O) Result (M) VotesFlorida 49.2% 48.0% 51.0% 48.2% 27Indiana 45.1% 50.4% 50.0% 48.9% 11Missouri 48.8% 48.8% 49.3% 49.4% 11North Carolina 49.1% 49.5% 49.7% 49.4% 15Ohio 49.4% 47.4% 51.5% 46.9% 20Virginia 51.7% 45.3% 52.6% 46.3% 13

(a) What is the margin of error for these Zogby poll results?(b) How many of the actual state election results were within the margin of error of the poll

results?(c) Write a short paragraph outlining several reasons why the poll results might have been

different than the election results. You should refer to relevant concepts discussed inthe text for at least some of these reasons.

(d) How many people would Zogby have to poll to reduce their margin of error to 2%?(e) Obama won the electoral vote tally 365 to 173. What would the tally have been if

McCain had won all six of these battleground states?

51Actual election results are available at www.cnn.com; the results we provide in the table are from wikipedia

4. Functions

If you think your teacher is tough, wait until you get a boss. He doesn’t have tenure.— Charles Sykes, author of Dumbing Down Our Kids

Every fall, students at thousands of other colleges and universities across the country pay fortuition. The amount each student is charged typically depends on the number of credit hourstaken. At Dakota Wesleyan University, in the fall of 2007, this was determined as follows52:

1-5 hours $375 per hour6 hours $2,525 flat rate7 hours $3,150 flat rate8 hours $3,850 flat rate9 hours $4,475 flat rate10 hours $6,100 flat rate11 hours $7,725 flat rate12-16 hours $8,750 flat rateOverload $375 per hour over 16 hours

If the number of credit hours are known, the tuition charge is exactly known. We say that thetuition charge is a function of the number of credit hours. The “number of credit hours” is calledthe independent variable, or the input. For a given student, it can vary anywhere from 1 upto as many as 21 or so (although there is no set upper limit, students need permission to registerfor more than 16 hours and 21 is the most the author has seen). The “tuition charge” is called thedependent variable, or the output, since how much a given student pays depends on how manycredit hours that student registers for. Which variable is independent and which is dependent issometimes given, but sometimes it will be your choice. If you are wondering how to choose, a goodidea is to just ask yourself “Does A depend on B, or does B depend on A?” If the answer is neither,then the choice is somewhat arbitrary.

Alternatively, we will sometimes say that the table above describes the function “tuition chargeversus number of credit hours,” instead of saying “the tuition charge is a function of the numberof credit hours.” When we describe a function by saying “A versus B,” the A is always thedependent variable and the B is the independent variable.

We will see many other examples of functions as we proceed through the course. Functionsare one of the most fundamental concepts of mathematics. They are very useful for describing(or what mathematicians and other scientists call modeling) all sorts of real-world situations andproblems.

Just What Is a Function?

We have already used the same “dependent variable versus independent variable” language whendiscussing two-variable data, especially scatterplots and other two variable graphs. We discussedhow scatterplots can represent negative or positive associations between variables. Functions area special kind of relationship between two variables.53

A function is a rule that, for each valid input, assigns one and only one output.

A function is a special type of a more general concept called a relation.

A relation is any set of ordered pairs. In each ordered pair (x, y) we consider x to bean input and y to be an output.

52This is, of course, not including any financial aid53Actually, more than two variables can be involved, as we will see in a few examples later

73

74

For example, the description of tuition policy at Dakota Wesleyan given above represents afunction where the input is the number of credit hours and the output is the (gross) tuition charge.For any positive number of credit hours we get exactly one tuition charge. If the input is 5, theassigned output is 5 × $375 = $1875. If the input is 17, representing a 1 hour overload over thestandard 12-16 hour load, the assigned output is $8, 750 + $375 = $9, 125.

A number of the examples considered in Chapters Two and Three represent functions. Forexample, the U.S. gasoline prices given graphically in Figure 4 and numerically in Table 1 ofchapter two represent a function where the input is the year and the output is the average gasolineprice. For each input year, there is a single average price given as the output.

An example of a relation which is not a function would be considering the inputs to be thenumber of credit hours taken by students during the fall semester at Dakota Wesleyan, but lettingthe outputs be the amount paid out of pocket for each student taking a given number of credithours. For example, two students who each take 12 hours might have different amounts paidbecause of differences in scholarships and other financial aid. For these two students, we mighthave ordered pairs like (12, $3, 500) and (12, $4, 200). The entire relation would consist of the setof ordered pairs for all Dakota Wesleyan students for the fall 2007 semester.

This relation is not a function because we do not have a single output assigned to eachinput. We will surely have two students with the same number of credit hours as theinput and two different amounts paid as the output.

Why is the distinction between functions and relations important?

Functions have the advantage over relations in that, for a function, knowing the input exactlydetermines the output. This is not true for a relation. This advantage is so useful that our standardapproach to modelling data will be to create functions which will approximate the data.

Four Ways to Represent a Function

In Chapter Two, we considered two ways to represent data, namely graphically and numerically.For functions, we will consider four possible representations, including these two.

In your previous mathematical experience, you are probably most accustomed to thinking offunctions as formulas given in symbols, as in y = x2. In this context, unless otherwise noted, yis the dependent variable or output and x is the independent variable or input.54 We will alsoconsider three other ways to represent functions. The four ways are:

1. Symbolically, as in y = x2.

2. Graphically.

3. Numerically.

4. Verbally, as in our tuition example above.

Symbolic Representations

As another example of a symbolic representation, consider

C = 2πr.

In this function, the input r represents the radius of a circle, and the output represents the cir-cumference C of the circle. The symbol π is not a variable, but instead is referred to as a constant;π ≈ 3.14159. For any input radius, the rule assigns a single circumference as the output since there

54The author is not sure of the origins of this practice, although it has been followed for many years.

4. Functions 75

is only one way to calculate the formula for any given input. If the input is r = 4, the output isC = 8π. There is no other possible output.

An example of a formula that does not represent a function is y = ±√x. Here, if the input is 4,the output could be either 2 or −2. Since the rule does not assign a single input for each output,the rule does not represent a function.

When determining if a rule represented symbolically is a function, you need to reason whetheror not any single input can lead to more than one output. There is no one procedure that youfollow – how you determine if the rule is a function will depend on the particular formula given.

Graphical Representations

We have already commented that the graph of U.S. gasoline prices given in Figure 4 of ChapterTwo represents a function. As another example, the World Population Growth Rates graph inFigure 1 of the Introduction also represents a function. For each input year, there is one growthrate depicted by the graph.

Typically, it is easy to determine if a rule represented graphically is a function by using thevertical line test. If there exists a vertical line that will cross the graph at more than one point,then there is more than one output for the input value where the vertical line crosses the horizontalaxis, and so the graph does not represent a function.

An example of a graph which does not represent a function is the graph in Figure 1, depictingthe relationship between Per Capita Income for each state versus the number of electoral votesfor each state. There are 538 total electoral votes, and every state gets at least 3 electoral votes.Based on the most recent census in 2000, California has the most electoral votes with 55.

Per Capita Income Vs Electoral Votes: 2000

0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

45,000

0 10 20 30 40 50 60

Electoral Votes

Figure 1: Do people in larger states make more money?

This graph is does not represent a function, since there are quite a number of states with 3electoral votes (input value 3), and these states have widely varying values for their per capitaincomes (output values). So, there is an input value which is not matched with a single output,but rather is matched with more than one output. You can probably find other input values besides3 which are matched with more than one output value.

As long as we have this graph handy, we might ask what this graph tells us about the relationshipbetween population and per capita income. You might recall that the number of electoral votesequals the number of representatives a state has in the House of Representatives plus two (sinceeach state has 2 senators). The number of representatives is based upon population, with each

76

representative representing very roughly 650,000 people.Notice that small states seem to display a larger variation in incomes. States with fewer than

10 electoral votes include both the poorest and the richest states. However, note that all the stateswith under $25,000 in per capita income are smaller states. To the extent that larger populationstates tend to have larger cities, this might be a result of there being higher per capita incomes inurban areas than in rural areas.

Numerical Representations

Checking if a table of data represents a function is fairly easy. You need to check whether aninput is paired with more than one output. If this occurs, then the table of data does not representa function. Table 1(a) represents a function since each input has exactly one corresponding output.However, in Table 1(b), the input 3 is paired with two different outputs (6 and 8), so this does notrepresent a function.

Table 1: The data in Table 1(a) represents a function while the data in Table 1(b) does not.

Input 1 3 4 6 9Output 3 6 8 6 5

Input 1 3 3 6 9Output 3 6 8 6 5

(a) (b)

Typically, when a function is given in numerical form as a horizontal table, the input variableis in the top row and the output variable is in the second row. If the data is in columns, the inputis typically in the first or lefthand column and the output in the second or righthand column.

What if there are more than two columns (or rows) of numbers or values in a table? In this case,it may be that the table describes several functions. For example, Table 2 shows median incomes(adjusted for inflation using 1998 dollars; more on this later) for six different groups of people.So for example, we could consider each of the income columns as outputs for one of six differentfunctions, all six functions having years as the input. One such function could be described as“Median Income of Black Women versus Year.” For this function, if the value of the input is 1955,the value of the output would be $3,658.

Table 2: U.S. Median Incomes in 1998 constant dollars

All Races All Races White White Black BlackYear Men Women Men Women Men Women1950 15,989 5,929 16,854 6,595 9,152 2,9491955 18,809 6,274 19,851 7,013 10,447 3,6581960 20,653 6,383 21,747 6,844 11,440 4,2371965 23,940 7,249 25,213 7,688 13,569 5,5951970 26,325 8,829 27,671 8,943 16,414 8,1421975 25,677 9,818 26,973 9,919 16,126 9,0111980 24,816 9,744 26,397 9,798 15,862 9,0711985 24,709 10,933 25,921 11,145 16,312 9,5091990 25,308 12,559 26,402 12,867 16,048 10,3861995 24,131 12,974 25,557 13,173 17,119 11,7231998 26,492 14,430 27,646 14,617 19,321 13,137

Table 3 gives another example of a function in numerical form.

4. Functions 77

Table 3: The probability distribution for rolling two dice and recording the sum.

Outcome: Sum 2 3 4 5 6 7 8 9 10 11 12Probability 1

36236

336

436

536

636

536

436

336

236

136

In this function, the inputs are the possible numbers, 2 through 12, that can be the sum whentwo dice are rolled. The outputs are the probabilities for the given sums. If the input is 4, theoutput is 3/36 or 1/12. This means the probability of getting a sum of 4 when rolling two dice is1 out of 12.

Verbal Representations

Our tuition example at the beginning of this chapter was one example of a function given in averbal representation.

As another example, we could describe a function verbally by saying

“Give the output as the number of square miles of land area of the given country”

or

“Tell me the square miles of land of the given country”

Each country is an input, and the output is the number of square miles of land in that country.Using our “versus” language, we could say that the function is “Square miles versus country.”

As a third example, “mother versus child” describes a function. Here, “child” is the independentvariable, and we define the mother as the biological mother. The identity of the mother dependson the child.

Note that for this function, a given person might serve as both an input and an output. If theauthor is the input, Marcella Catalano is the output. However, Marcella can also be an input andthe output would be Anna (Trnka) Herda. There are lots of functions where particular values canserve as either inputs or outputs.

On the other hand, the rule “child versus mother” is not a function. If mother is the inputvariable, there will very often be several outputs for a given input (anytime a mother has morethan one child). For example, if the input is Marcella Catalano, there are four outputs, namelyMichael, Gabriela, Aaron, and Naomi.

Sometimes, one needs to be careful in determining exactly what the input and output variablesare. For example, for the rule

“Tell me the weight of a person who eats a given number of calories per day, on average.”

The output is weight, but the input is not the person but the number of calories the personeats. The person is just part of the rule which associated the calories with the weight. This ruleis not a function, since two different people could eat the same number of calories per day (sameinput) but have different weights (outputs).

If we changed the rule to say

“Tell me the weight of a given person.”

then this is a function, since now the input is the actual person, and each person has only oneweight (at any given time).

If we have an input and output variable combination that does not give a function, we oftencall it a relation. Any table or any graph in rectangular coordinates represents a relation. Thus,we can think of a function as a relation with the additional condition that each input correspondsto exactly one output.

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One reason we use functions in verbal form is that it allows us to consider the dependence ofone quantity (the output) on another (the input) before we have a symbolic representation or anynumerical information.

Why All These Representations?

We will use all four representations of functions – symbolic, graphical, numerical, and verbal– throughout this book. All four representations are equally valid. There is not a “right” or“wrong” way to represent a function. Rather, there are four equally valid ways of representing thesame thing. However, one of the four representations might work best for a particular situation,depending on what the function represents and what you intend to do with it. Consider thefunctions represented in Table 2 above, where the input is the year and the outputs are medianincomes for various groups. To quickly communicate the number of people in a given year, thetable probably works the best. If you want to illustrate how incomes are changing, a graphicalrepresentation would be better. If you are interested in describing these functions to someone else,you would probably give a verbal description. However, if you want to use a function to predictincomes five or ten years from now, you will probably want to find a symbolic representation. Sowhile all four representations are equally valid, you may prefer one representation over another toconvey certain information.

Function Notation

Functions in symbolic form are often denoted using the so-called function notation. In thisnotation, we let f(x) (pronounced as “f of x”) denote the output where x denotes the input. Thef(x) is a generic notation, and for a particular function, we give an explicit expression for f(x).The letter f by itself can be thought of as a label representing the function, while f(x) technicallyonly represents the output of the function.

For example, f(x) = 3x+2 means “my function, called f , is the rule that multiplies a given inputby 3 and then adds 2.” There is nothing special about the letter x. It should be thought of as a place-holder or “blank” for your input. So f(x) = 3x + 2 should be thought of as f( ) = 3( ) + 2.For example, f(4) = 3 · 4+2 = 14, f(−1) = 3(−1)+2 = −1, f(π) = 3π +2. In fact, f(y) = 3y +2and f(x + 1) = 3(x + 1) + 2, which we can simplify to 3x + 5.

When a function is a modeling function, it is customary to use letters which represent the givenquantities. For example, the function for the area of a circle is typically written as A = πr2 ratherthan f(x) = πx2 to remind us that the input is the radius of the circle and the output is the Areaof the circle.

Notational Issues

A couple of notes on function notation and terminology. When we write f(x), remember f(x)does not mean f times x. The f by itself is not even a number but rather is the generic designationfor a function. The expression f(x) will sometimes be referred to as the value of the function f atthe input value x. So, for example, if f(x) = 3x + 2, the value of f at 6 is equal to f(6) or, usingthe formula, 20.

Here is an analogy that might be helpful. Let’s let f stand for the function which representsthe instructions for filling out a 2007 tax return to get the amount of tax owed for the 2007 taxyear, and let x represent any given tax payer. Then f(x) represents the amount of tax owed bytaxpayer x for 2007. Notice that f by itself does not have any monetary value. You cannot findthe tax owed unless you know what particular tax payer you are talking about. The letter f ismerely the name for the rule (a very elaborate rule with dozens of pages of instructions) whichtells you how to find the value of the tax owed for each taxpayer x. If you are a taxpayer, f(you)is the amount you owe for 2007.

Also, we will often write the equation y = f(x). In this case, we are simply using a single letter,namely y, to denote the output values, rather than the expression f(x). This is sort of like callingthe President “Mr. Bush” instead of “President George W. Bush.”

4. Functions 79

US Flood Damages Adjusted for Inflation

$0.00

$4.00

$8.00

$12.00

$16.00

$20.00

0 20 40 60 80 100

Year from 1900

Dam

ages

in B

illio

ns

of

$

Figure 2: Two functional representations of US Flood Damages.

Modeling Functions

Many of the functions we will consider throughout this text will be examples of modeling func-tions. A modeling function is a function that describes a real world phenomenon by representingthe relationship between two real world variables. For example, y = x2 is a function which givesthe square of the input. However, if this function is used to find the area of a square, then y = x2

(or A = s2) is now a modeling function because it represents the area of a square as a function ofthe length of the side. We might also specify the units for s, say in feet. The units for A wouldthen be square feet.

Modeling functions can be represented in all four ways – symbolic, graphical, numerical, verbalor by combinations of these. However, if we wish to use a modeling function to predict somethingwhich has not yet occurred, to calculate an output value which is difficult to estimate in anotherway, or to expand the list of input output pairs in a table, a symbolic representation is often themost useful.

For example, global warming is a topic that has been in the news a great deal of late, andis likely to remain so into the foreseeable future. Global warming has many potential effects,including more frequent and more intense hurricanes with resulting damage caused by wind, rain,and flooding. Suppose we wish to use a function to model flood damage in the U.S. The graph inFigure 2 actually shows two functions. The first, represented by the “squiggly line graph,” showsactual flood damages in billions of dollars adjusted for inflation. The second “straight line” is alinear approximation of the first.55

Both graphs give us a picture of how flood damages have been changing over time. In general,flood damages have been increasing, albeit with a lot of variability from year to year representedby the squiggles.

In order to predict what might happen in the future, a symbolic representation might be useful.In fact, the linear approximation graphed above has the symbolic representation

D(t) = 0.036t + 1.25

55Data from U.S. National Oceanic and Atomspheric Administration (NOAA) http ://www.nws.noaa.gov/oh/hic/floodstats/F loodlosstimeseries.htm.

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where D(t) is the approximate damages in billions of dollars and t is the time in number of yearssince 1900.56 Notice we are using the letter D to denote our Damages function rather than themore generic f . This is not necessary, but can be helpful, especially when one is considering severaldifferent modeling functions simultaneously.

Using this symbolic representation, we can estimate flood damages in future years. For example,for the year 2010, t would be equal to 110. The estimated damages would be

D(110) = 0.036 · 110 + 1.25 ≈ 5.21.

Going further into the future, the estimated damages for 2030 would be D(130) = 0.036·130+1.25 ≈5.93.

Now, we should make a couple of comments about these predictions. The first is that, becauseour symbolic representation is only an approximation and we know that the actual damages showsa high degree of variability, we should expect that our estimate could be wrong by several billionsof dollars. Secondly, the model assumes that no changes in the behavior of the system that mightsignificantly effect the dependent variable occurs in the future. For example, if ocean temperaturesbegin to increase more rapidly in the future, this may lead to even more frequent and strongerhurricanes, with increased resulting flood damages.

Example 1. For the function D modelling U.S. flood damages, explain what D(60) = 3.41means in the given context.

Solution: Since D is giving an estimate of U.S. flood damages adjusted for inflation where theinput is years from 1900, D(60) = 3.41 means we estimate that U.S. flood damages were about3.41 billion dollars in 1960. We will note that actual damages were about 0.69 billion. Again, thisis mostly because our symbolic calculation does not give actual flood damages but only estimateddamages.

Example 2. We noted earlier that the graph of fertility versus contraceptive use in Figure 1 ofthis chapter probably does not represent a function. However, the line that is shown along withthe scatterplot does define a function. One way to see this is that any vertical line will intersectany other line (except another vertical line) in exactly one point. While we do not have a symbolicrepresentation for this function, like the function D given above which models U.S. flood damages,this line was created to be the best linear model for this data. Having only the graph of the line,we could still use the line to make “eyeball” estimates of “average” fertility rates as a function ofcontraceptive use. For example, following the line, a country with a contraceptive use rate of 60%would average about 3 births per woman.

Example 3. We are certainly allowed to consider different functions as models of the same dataor phenomenon. Another model for U.S. flood damages is given by

D1(t) = 0.741(1.0168t)

where we have used D1 to distinguish this model from the previous one. What does this secondmodel predict for flood damages in 2010 and 2030?

Solution: The estimates provided by D1 are D1(110) = 0.741(1.0168110) ≈ 4.63 and D1(130) =0.741(1.0168130) ≈ 6.46. Notice that D1 gives a lower estimate than D for 2010 but a higherestimate for 2030.

Whether D1 is a better model than D is a question we will consider further in later chapters.

Composition of Functions

Previously, in our “mother versus child” function, we noted how a given value, like MarcellaCatalano, can serve as either an input or an output. Similarly, using function notation, a given

56This equation was derived using linear regression – something we will learn about later in this book. The givenpoints will not necessarily lie on the line and, therefore, will not necessarily be solutions to the equation.

4. Functions 81

output g(x) can serve as the input for another function f . For example, if g is the function whichassigns to each person his or her mother, and f is the function which assigns to each person his orher father, then for a given person x we can first find the person’s mother g(x) and then use thisas the input in the function f , finding f(g(x)). The expression f(g(x)) represents both the fatherof g(x) and also the maternal grandfather of x.

Example 4. Let g be the “mother function” and f be the “father function” as described above.Explain what g(f(x)), f(f(x)), and g(g(x)) mean.

Solution: The expression g(f(x) signifies that we first find the father f(x) of person x, andthen use f(x) as the input in the function g(x) to find the mother of f(x). The whole expressiong(f(x)) thus represents the mother of x’s father, or in other words, x’s paternal grandmother.

Similarly, f(f(x)) represents x’s paternal grandfather and g(g(x)) represents x’s maternalgrandmother.

The process of using the output g(x) of the function g as the input for the function f is calledthe composition of f and g. We use the notation f ◦ g to denote the composition of f and g.Thus, (f ◦ g)(x) or f(g(x)) denotes the value of the composition of f and g.

Expressing the composition f ◦ g symbolically using the symbolic expressions for f and g isfairly straightforward. For example, if f(x) = 2x + 5 and g(x) = 3x− 1 then

(f ◦ g)(x) = f(g(x)) = f(3x− 1) = 2(3x− 1) + 5.

If we wish, we can simplify the last expression using the distributive law, so that

(f ◦ g)(x) = 2(3x− 1) + 5 = 6x− 2 + 5 = 6x + 3.

Note that what we are doing is merely substituting the expression 3x−1 for the x in the expressionfor f(x). Alternatively, we could first substitute the explicit expression for f , then substitute forg, and still get the same expression.

(f ◦ g)(x) = f(g(x)) = 2(g(x)) + 5 = 2(3x− 1) + 5

Example 5.Let f(x) = x2 + 3 and g(x) = x− 2. Find (f ◦ g)(x) and (g ◦ f)(x).

Solution: We have

(f ◦ g)(x) = f(g(x)) = f(x− 2) = (x− 2)2 + 3 = x2 − 4x + 4 + 3 = x2 − 4x + 7

and(g ◦ f)(x) = g(f(x)) = g(x2 + 3) = x2 + 3− 2 = x2 + 1.

Note well from the previous example that (g ◦ f)(x) and (f ◦ g)(x) give you different formulas,and thus are different functions. This will be the case for the vast majority of pairs of functions fand g.

Domains and Ranges

Sometimes, we may wish to talk about the whole set of possible input values, or the whole setof possible output values, rather than individual inputs or outputs.

• The set of all possible inputs for a given function f is called the domain of f .• The set of all possible outputs for a given function f is called the range of f .

82

For example, if we are considering the function “Tell me the weight of the given person,” thenthe domain is the set of all people, and the range is the set of all weights of people. If we aremeasuring weights in pounds, we might say the range is all number bigger than 0 (prematurebabies can weigh only a few ounces) up to the largest weight ever recorded (say 1200 pounds as aguess, or consult the Guiness Book of World Records for a more precise upper limit).

For a function given in numerical form, the domain would be simply the list of inputs, unlessit is understood that the function includes more inputs and outputs than are listed. For example,for the functions given in Table 2 on median incomes in the U.S., we might say the domain is thelist of years 1950, 1955, 1960, up to 1995 and 1998. Similarly, the range of a function representedby one of the income columns would simply be the list of 11 incomes. For example, the function“Median Income for Black Women versus Year” has range

{$2949, $3658, $4237, $5595, $8142, $9011, $9071, $9509, $10386, $11723, $13137}.For a function given in symbolic form, we usually take the domain to be the largest set of real

numbers which are valid when substituted into the formula for the function. For example, if thefunction output is given as f(x) = 1/x, then the only input that would not work of x would bex = 0, since division by 0 is not defined. Thus, we would say the domain is “the set of all realnumbers except 0.”

For now, the two main considerations when considering domains of symbolic functions are to:

• Exclude all input values which result in division by 0 from the domain.

• Exclude from the domain all input values which result in taking the square root of a negativenumber.

For example, if f(x) =√

x−2x−5 , then we would exclude all numbers smaller than 2 from the

domain, since these numbers would result in taking the square root of a negative number. Wewould also exclude 5, since that would result in division by 0. So, the domain would be “all realnumbers bigger than or equal to 2 and not equal to 5.” In mathematical notation, we might writethis as {x|x ≥ 2, x 6= 5}.

If a function given in symbolic form is also a modeling function, we might further restrict thedomain and range based on the context. For example, for the function y = x2, we could say thedomain is all real numbers, since there is not restriction on what x might be for this equation. Thefunction A = s2, where s is the side length of a square and A is the area, is given by essentiallythe same equation. However, it does not make sense for s to be 0 or negative in this context, sowe would say the domain is all positive real numbers. In general, for modeling functions we simplypick a reasonable set of values for the domain, based on the context, and allow some flexibility onwhat counts as “reasonable.”

Finally, for functions in graphical form, we can often infer the domain and range off of thegraph. For example, for the line shown in Figure 5, we might say the domain is “all percentagevalues from 0% to about 90%,” based on where the line starts and ends. We could say the rangeis approximately “all real numbers from about 1 up to about 6.2.”

Functions and solving equations

Previously, we used the function D(t) = 0.036t+1.25 to model U.S. flood damages as a functionof the year since 1900, denoted by t. We estimated the flood damages in 2010 by evaluating D(t).What if we want to know when flood damages will reach $7 billion or some other amount?

Here, we are asking what input t would produce an output D(t) equal to 7. Since D(t) =0.036t + 1.25, this results in the equation

7 = 0.036t + 1.25.

You very likely have solved equations like this before, even if they were not in the context of workingwith a function. The basic rules are:

4. Functions 83

• Equals can be added to both sides of any equation.

• Equals can be subtracted from both sides of any equation.

• Equals, except for zero, can be multiplied on both sides of any equation.

• Equals, except for zero, can be divided by on both sides of any equation.

Following any of these rules changes a given equation into a different equation, but the secondequation will still have the same solutions(s) as the first.

The “except for zero” is important in the last two items. For example, the equation x− 2 = 5has one solution for x, namely 7. However, if we multiply both sides by zero, we have

0(x− 2) = 0 · 7

0 = 0

This last equation is trivially true, and does not even have the variable x appearing. Thus, it istrue no matter what the value of x is. So, the second equation does not have the same solutionsas the first. This is a result of multiplying by zero, and thus, is why multiplying both sides of anequation by zero is not a legitimate step in solving equations.

Now, to solve 7 = 0.036t + 1.25, we first subtract 1.25 from each side of the equation.

7− 1.25 = 0.036t + 1.25− 1.25

5.75 = 0.036t.

Then we divide both sides by 0.036 (which is not zero).

5.750.036

=0.036t

0.036

159.7 ≈ t

If we round the value t = 159.7 to the nearest year, we get 160, which corresponds to the year 2160.So, our equation would predict U.S. flood losses should reach $7 billion (on the average) around2160.

Example 6. For the function f(x) = 2x + 5, find the input x when the output f(x) is 3.

Solution: We have

3 = 2x + 5−2 = 2x−1 = x

Example 7. Consider the function C(r) = 2πr, where C(r) represents the circumference of a circle

as a function of the radius r.

• Find the circumference of a circle of radius 2.5.

• Find the radius of a circle which has a circumference of 220.

84

Solution: If the radius is 2.5, then the circumference is equal to C(2.5) = 2π2.5 = 5π ≈ 15.71.

Now, if the circumference is 220, this means C(r) = 220. So, we have

220 = 2πr

220/(2/pi) = r

r = 110/pi ≈ 35

So, if the circumference is 220, the radius is approximately 35.

Characteristics of Graphs of Functions

In this section, we introduce some terminology we can use to describe the shape of the graphof a function. These terms generally apply to graphs which are in the form of a line graph, versusa scatterplot.

Increasing and DecreasingA function is said to be increasing over an interval if, for each input, a, in the interval, every

input to the right of a has a greater output.57 Symbolically, we say a function is increasing onan interval if for any two numbers a and b in the interval with b > a, then f(b) > f(a). In words,this says the larger the input, the larger the output. Increasing graphs are going “uphill” as xincreases toward the right. This does not have to occur over the entire graph since increasing is aproperty on an interval. In our world oil production example (Figure 2), the graph is increasingon the interval from 1930 to about 2010, except for a short interval in the early 1970’s, and asomewhat longer interval from 1979 to about 1982.

A graph is decreasing over an interval if for each input, a, in that interval, every input to theright of a has a smaller output. Symbolically, we say a function is decreasing on an interval iffor any two numbers a and b on the interval with b > a, then f(b) < f(a). In words, this says thelarger the input, the smaller the output. Decreasing graphs are going “downhill” as x increasestoward the right. The graph in Figure 2 is decreasing from 2010 on. To determine if a graph isincreasing or decreasing, we always compare outputs as we move from left to right.

Steepness is another word we can use to describe a graph or a section of a graph. The morequickly a graph increases or decreases, the steeper it is. We can think of steepness as indicatingthe rate of change of the graph, and will say more about this later.

One-to-one Functions

Previously, we discussed the vertical line test for functions. Any graph that is intersected atmost once by any vertical line is a function. We can also consider a horizontal line test for graphsthat we already know are functions. If the graph of a function is intersected at most once by anyhorizontal line, then we say the function is one-to-one. Verbally, this means that each output ismatched with only one input. This is the converse of the definition of a function which states thateach input is matched with exactly one output.

Symbolically, a function f is one-to-one if the only way that f(a) = f(b) is if a = b. So, forexample, the function g(x) = x2 is not one-to-one since g(3) = g(−3), but 3 6= −3. Functions thatare one-to-one include those that would be increasing through their entire domain, or those thatwould be decreasing through their entire domain. Both of these types of functions would satisfythe horizontal line test.

ConcavityIn Chapter Two, we discussed line graphs as one type of graph. For line graphs that are

the graphs of functions, we can categorize the graph or sections of the graph according to theirconcavity. Informally, a graph is concave up if it bends upward, resembling a bowl that can

57The interval must be contained in the domain of the function.

4. Functions 85

hold water. A graph is concave down if it bends downward, resembling an umbrella when itsheds water. The concavity of a graph and whether it is increasing or decreasing over an intervalare independent. An increasing function can either be concave up or concave down. Similarly, adecreasing function can be either concave up or concave down. (See Figure 3.)

Increasing and concave up

Increasing and concave down

Decreasing and concave up

Decreasing and concave down

Figure 3: Examples of concavity

–4

–3

–2

–1

0

1

2

3

4

y

–3 –2 –1 1 2 3

x

Figure 4: The graph of y = 1/x is always decreasing, is concave down when x < 0 and concave upwhen x > 0.

One needs to be careful in distinguishing between the concepts of increasing/decreasing, concaveup and concave down, and slope and steepness. Increasing and decreasing describe how the outputof the function is changing over an interval while the concavity of a function tells how the rate ofincrease or decrease is changing on the interval. This is connected to the definition of concavity.A graph is concave up on intervals where the rate of change or slope of the graph increases.A graph is concave down on intervals where the rate of change decreases. Note that when thegraph is increasing, we say the slope is positive, and when it is decreasing the slope is negative.

Also, we need to be careful that slope is not the same as steepness. If a graph is increasingand concave up, the slope of the graph is increasing and we also would say it is getting steeper.On the other hand, if a graph is decreasing and concave up, then its slopes are negative but arestill increasing. The slope becomes “less negative.” However, the graph is also getting less steepor “flatter.” One needs to carefully consider this difference between steepness and slope and howit is affected by whether the graph is increasing or decreasing.

For example, the graph in Figure 4 is always decreasing. Where it is concave down (when

86

x < 0), its rate of change is decreasing, and in this case it is becoming steeper. The slope isbecoming more and more negative.

Where it is concave up (when x > 0), the slope is still increasing (becoming less negative), butit is becoming less steep. In other words, its rate of change is increasing, even though its steepnessis decreasing.

Example 8. The graph of y = 1/x2 is shown in Figure 5. Describe where this function isdecreasing, increasing, concave up, and concave down.

–2

–1

1

2

3

4

y

–3 –2 –1 1 2 3

x

Figure 5

Solution: The function is increasing when x < 0 and decreasing when x > 0. It is alwaysconcave up and never concave down, so the rate of change of the function is always increasing.

Functions Related to Hunger

Every gun that is made, every warship launched, every rocket fired, signifies in the finalsense a theft from those who hunger and are not fed, those who are cold and are notclothed.—Dwight D. Eisenhower

How can one “measure” hunger?

Hunger is not a numerical quantity. However, we could define and try to measure it in a numberof ways. For example, we might define hunger to mean not getting the minimum number of caloriesneeded to function normally on a day to day basis. As long as we can measure a person’s calorieintake we can measure hunger, under this definition. We might consider a person hungry if theyoften have to skip meals, or go days at a time without eating. We can measure this by countingmeals skipped, assuming 3 meals a day as normal. Alternatively, we might be more concernedwith the long term effects on people who are chronically hungry, including mortality that can beattributed to hunger.

These measures all can be applied to individual people, but we may also want to developmeasures which apply to countries, social groups, or other populations of people. For example,UNICEF has three different standard ways of measuring malnutrition within countries. Theseare the percentage of children underweight for their age, the percentage of children suffering fromstunting, and the percentage of children suffering from wasting. Stunting is a measure of height

4. Functions 87

deficiency, and can result from chronic hunger over months or years during a child’s growth period.Wasting results when a child is underweight for their height, in other words, chronically thin. TheUNICEF data set including these malnutrition measures can be found in the appendices, and wewill use these data in a number of examples, exercises, and activities throughout this book.

Of course, we want to do more than just measure hunger. Ideally, we want to identify, ifpossible, what causes hunger, so that we can address the problem constructively. This mightinclude identifying variables which we can numerically measure, and seeing how closely theseare correlated with our measures of hunger.58 It would be unrealistic to expect that we couldfind variables X for which “hunger is a function of X.” However, we should certainly be able tocreate functions, in symbolic or other representations, which serve as modeling functions for therelationship between hunger and other variables.


0

5

10

15

20

0 10 20 30 40 50 60 70 80

National Poverty Rate (as percent)

Per

cen

t S

ever

ely

Uw

t C

hild

ren

Figure 6: Percentage of Severely Underweight Children versus National Poverty Rate

For example, Figure 6 shows a scatterplot of malnutrition versus poverty, where malnutritionis measured using the underweight measurement. Poverty is measured for each country by itsnational poverty rate, which is the percentage of the population considered below the nation-adjusted poverty line. What does this graph seem to say about the relationship between povertyand hunger?

In the activities for this chapter, you may be asked to identify other variables which might berelated to hunger.

58Correlation will be defined in chapter 6. For now, we will note that two variables are said to be correlated if anumerical change in one tends to produce a change in the second.

Reading Questions for Functions

1. Name the four different ways to represent a function, and give examples of each. [Note:The examples should be different from those given in the reading.]

2. For each of the following, determine if the rule represents a function. Write a sentencejustifying your answer.

(a) Tell me the date of birth of a given person.(b) Tell me the total amount owed in student loans of a given graduating senior.(c) Tell me the number of calories you consumed on a given day.(d) Tell me a color on the flag of a given country.(e) Total yearly income versus number of years of education attained by given person.(f) Number of malnourished children in a country versus number of personal vehicles in the

country.(g) Tell me the grade of a given person in College Algebra this semester.(h) Tell me the grade of a given person in College Algebra where the input is how many

hours they study per week for the course, on average.

3. For each of the following, determine if the rule represents a function. Write a sentencejustifying your answer.

(a)Input −4 −1 0 2 5

Output 3 4 6 6 9

(b) y = x + 3, where x represents the input.(c) y2 = x2, where x represents the input.(d) y2 = x + 2 where x represents the input.(e) y2 = x + 2 where y represents the input.

(f)Input 3 −1 0 3 5

Output 7 2 5 5 3

4. For each of the following modeling functions, give reasonable values for the domain and alsofor the range.

(a) w(h) = 1.4h − 20 where h is the height in inches of a child under 3 years of age, andw(h) is the estimated weight in pounds of the child.

(b) C = 2.959g where C is total cost of gasoline and g is the number of gallons purchasedfilling up a single gas tank.

(c) The cost of tuition at your college in a given year.(d) The number of children a particular woman has as a function of her age (as their age

varies over their lifetime).(e) The total yearly income of a particular person as a function of the age of that person.

5. In the previous problem, what does the 2.959 in part (b) represent?

6. What is typically the best way to represent a modeling function if you want to use it topredict something that has not yet occurred?

7. Use the symbolic representation D(t) = 0.036t + 1.25 of the function from this chapter thatmodeled U.S. flood damages to estimate flood damages for the year 2000. Remember that tis the number of years since 1900. Use the symbolic representation of the function D(t) toestimate flood damages in 1850. Does your answer seem reasonable? Why or why not?

88

4. Functions 89

8. The following table gives the estimated percentage of all Americans who had ever used mar-ijuana in the given year.

Year 1979 1982 1985 1988 1991 1993 1994 1996 2001Marijuana Use 28.0% 28.60% 29.4% 30.6% 30.5% 31.0% 31.1% 32.0% 36.9%

(a) What row is associated with the range?(b) How would you describe the range as a set?

9. When finding the domain for a function given symbolically, what two things should youcheck?

10. (a) What are the domain and the range of the function f(x) = πx2?(b) What are the domain and the range of the function A = πr2 used for finding the area

A of a circle, given the radius r?

11. Give an appropriate domain for each of the following functions. You may use symbols, words,or both when giving your answer.

(a) f(x) = 5x2 − 2

(b) f(x) =√

x + 6

(c) f(x) =√

4x− 3

(d) f(x) = 1x−8

(e)Input 0 2 5 8

Output −1 1 2 1

(f) Your summer job pays an hourly wage of $7.25 an hour. Given the number of hours youworked in one week, determine your gross pay for that week.

12. The cost C in cents of first class postage is a function of the weight w (in ounces) of the itemmailed. We write C = f(w). Explain the meaning of f(2) = 0.60.

13. Suppose that, in a particular state, the amount of the fine you pay for speeding F is afunction of the number of miles per hour over the speed limit you were clocked at, denotedm. We write F = f(m) (mathematics is generally case sensitive). Explain the meaning off(12) = $75.

14. Let f(x) = 3x2 + 4. Find each of the following:

(a) f(0)(b) f(2)(c) f(a)(d) f(x + 1)

15. Let f(x) =√

x−32x+1 . Find (where possible) each of the following:

(a) f(0)(b) f(4)(c) f(7)(d) f(x + 1), and simplify

90

16. Let f(x) = 3x + 4. Solve each of the following equations:

(a) f(x) = 7(b) f(x) = 0(c) f(x) = 11(d) f(x + 1) = 11(e) f(x) = x + 10

17. Let f(x) = 3x2 + 4. Solve each of the following equations. You may need to use the ‘guessand check’ method.

(a) f(x) = 7(b) f(x) = 0(c) f(x) = 16(d) f(x) = 22

18. Let f(x) = 3x + 4 and g(x) = 5x− 2. Find (f ◦ g)(x) and (g ◦ f)(x) and simplify.

19. Let f(x) = 2x + 3 and g(x) = 12x− 6. Find (f ◦ g)(x) and (g ◦ f)(x) and simplify.

20. Let f(x) = x2 + 4 and g(x) = 2x− 5. Find (f ◦ g)(x) and (g ◦ f)(x) and simplify.

21. Let f(x) = 3x2 + 4 and g(x) = 2x− 5. Find (f ◦ g)(x) and (g ◦ f)(x) and simplify.

22. Suppose the function m(n) = 2√

.25n gives the upper limit on the margin of error for a

randomly conducted poll of n people

(a) Find m(400) and m(1600).(b) Carefully explain what m(1100) = .03 in terms of the number of people polled and the

margin of error.(c) What sample size would be required to achieve a margin of error of .02?

23. A function approximating the data on marijuana use given previously in the Reading Ques-tions is y = 0.0033x+ .2414, where x is the year from 1970 and y is the estimated percentageof all Americans who have used marijuana, as a decimal.

(a) What is the estimated percentage use in 1970?(b) According to the equation, in what year did marijuana use reach 26%?(c) According to the equation, in what year will marijuana use reach 40%?(d) Would this equation eventually predict 100% marijuana use? What does this imply

about the applicability of this equation?

24. Graph the function f(x) = x2 − 2x + 1 on a computer or graphing calculator.

(a) Where is the function increasing?(b) Where is the function decreasing?(c) Where is the function concave up?(d) Is this function one-to-one or not?

4. Functions 91

Functions: Activities and Class Exercises

1. Midges.59 In this activity, you will be introduced to the Fathom Dynamic Data softwarepackage, and gain some experience working with functions in numerical and graphical form.

(a) A good way to look at all of the data at once is to use a case table. To open a casetable in Fathom, first select the collection by clicking on it. Then, you can either dragthe Table icon into the main window, or choose New—Case Table from the Objectmenu. Do this.

i. How many Af and how many Apf midges are represented in our data set? (You canrecord your answer in the same text box. Resize or move the box as needed).

ii. To find the longest and shortest wing span for this set of midges, you can select thetable and then the WL column, and then Table—Sort Ascending. Record thelongest and shortest wing spans in your text box. Note that Apf midges tend tohave longer wings than the Af midges.

iii. Do the same for Antenna length. Which variety of midges seems to have longerantennae?

iv. You can also use functions or formulas in symbolic form in Fathom. Let’s addanother attribute and create it as a formula. To do this, click on the new at thetop of the rightmost column of the table and then type in “Difference” which willbe our name for the new attribute. Then, click on Table—Show Formulas. Anew blank row should appear above the first case. Now, click on the box in this rowjust under your Difference heading. You should get a “formula dialog box.” Let’senter in the formula WL - AL. You can do this by typing or by selecting the WLand AL from the attribute menu in the formula box. Click OK when done. Thedifferences between WL and AL are calculated and entered into the table for eachmidge.

v. sort your table using the Difference column. What do you notice? What does thedifference tell you about the variety of midge?

(b) Create a graph of AL versus WL.

(c) Looking at either the table or the graph, decide whether or not AL is a function of WL,and explain why, using the definition of function.

(d) (Optional for those using tablet PC’s) You can include Fathom Documents in otherwindows documents, including Microsoft Journal notes. To do this, open the journalwriter from the START menu. Then, select File—Import. Then, select the Fathomfile. You should get a copy of your Fathom file to appear on the journal page. However,it may be that some of the Fathom screen got ’cut off’. If so, try again after deletingthe note (use File menu) and moving the objects in the Fathom window to the left, sothey will all fit in the journal window.

2. Hunger. The graph below shows a scatterplot of Child Mortality versus Percent UnderweightChildren. Child mortality is measured in number deaths of children under five per 1000 livebirths. There are 80 data points, representing 80 different countries. The data are from theyear 2000 and was compiled by UNICEF.60

(a) Why is the scatterplot probably not the graph of a function?

(b) Explain why the line graph shown with the scatterplot is a function.

59This activity uses data provided by Key Curriculum Press with the Fathom software60see /www.childinfo.org

92 Children Under Five Mortality vs Percent Underweight0501001502002503003504000 10 20 30 40 50 60

(c) The line shown is meant to represent the upper limit on child mortality given the percentof underweight children. Note that only two countries have points over this line. Theequation for this function is f(x) = 8.5x + 20, where x represents percent underweightchildren and f(x) represents the approximate upper limit on the child mortality rate fora country with the given underweight percentage x.

i. Find f(12), and explain briefly what this value means.ii. Find f(18), and f(24).iii. Suppose a country with a child mortality rate of 120 deaths per 1000 live births

wants to reduce its child mortality rate to no greater than 80 deaths per 1000 livebirths. Use the symbolic representation for f(x) to estimate what the percent ofunderweight children should be for the country to achieve this goal.

(d) So far, we have been considering the variable “Childhood Mortality” as it is related tohunger, where the latter is measured using “Percent of Underweight Children.” Whatother variables do you think might be related to hunger? Identify as many possibilitiesas you can. Variables might include those which you believe are causes of hunger, butyou could also include variables which you believe might be correlated with either hungera “lack of hunger.” Some of your variables might be numerical, but do not necessarilyrestrict yourself to variables which are easily measurable. If you possibly can, however,identify how the variable might be measured.

3. Education Spending. The following table gives the average amount of money spent perstudent in the United States for a number of years, going back to 1960, adjusted for inflation.The year represents the year in which that school year was completed (so 1960 represents the1959-60 school year).

Year 1960 1970 1980 1981 1986 1990 1991Spending $2,161 $3,657 $4,954 $4,890 $5,843 $6,639 $6,647Year 1994 1995 1996 1997 1998 1999 2000Spending $6,678 $6,741 $6,735 $6,811 $6,987 $7,216 $7,392

(a) Does this table represent a function? Why or why not?(b) One can quickly see from the table that spending, even after adjusting for inflation,

has increased over the years. We might wonder during which period of years educationspending was increasing the fastest. To do this, we might consider the percentage

4. Functions 93

increase in spending from one year to the next. One problem with our data would bethat we do not have spending values for every year.

i. Describe how would you calculate the percentage increase from one year to the next.ii. Estimate the percentage increase from 1990 to 1991.iii. How could you calculate yearly percentage increases for periods where you do not

have data for every year?iv. Use your idea from the previous question to estimate the yearly percentage increase

from 1960 to 1961. Using your method, would you get the same answer for all theyearly increases for the 1960’s?

4. How much does your car contribute to global warming? Based on a sample of 38 carmodels, the function y = G(m) = −.267m + 13.48 approximates the tons of carbon dioxideemissions G(m) produced by a car in a year as a function of the EPA’s city mileage ratingm for that car in miles per gallon.

(a) Find the value for G(20) and describe what it means in terms of gas mileage and carbondioxide emissions.

(b) Find the estimated carbon dioxide emissions for a car model that gets 30 miles pergallon.

(c) Find the estimated gas mileage for a car model which produces 7 tons of carbon dioxideannually.

(d) According to the given function, what is the estimated gas mileage for a car whichproduces no carbon dioxide emissions?

(e) Suppose that m(w) denotes a function which models the city mileage m(w) as a functionof the weight of the vehicle w in pounds.

i. Suppose m(1300) = 27. Explain what this means in terms of weight and mileage.

ii. Suppose m(w) = 9000.82w . Find an expression for the composition (G ◦m)(w).

iii. Identify the input and output variables of the function G ◦m and explain what thisfunction means in terms of these variable.

5. World Population Growth Rates. World overpopulation is frequently cited as a con-tributing factor to world hunger. The world population has grown from about 4,434,700,000in 1980 to about 6,336,200,000 in the latter part of 2003. However, the annual growth rateof the population has been decreasing for a number of years, and in 2003 was about 1.16%per year. This means that in 2004, there would be about 1.16% more people than there werein 2003. The graph below shows the percent growth rate for world population, going back to1950 and projected out to 2050.

(a) Given the estimated population in 2003 noted above, and the estimated 1.16% growthrate, about what would the population be in 2004?

(b) We could define a function verbally by letting the input be the year, starting with 2003,and the output be estimated world population assuming that the 1.16% holds foreverand the 2003 population is 6,336,200,000. You have already calculated the value for thisfunction when the input is 2004. Begin to develop a numerical representation for thisfunction by filling in the second column of the table below, giving the values for thefunction in the given years.

94


Est. Pop. at 1.16% Est. Pop. atYear annual growth Est. growth rates Est. growth rates2003 6,336,200,000 1.16 6,336,200,0002004 1.142005 1.122006 1.112007 1.102008 1.082009 1.072010

(c) Since the world population growth rates are decreasing, the estimated population func-tion from part (b) probably overestimates what the actual population will be (althoughwe won’t know this for awhile!). To get a better estimate, we could use the functionwhere the output is calculated using the estimated growth rates, as they change overtime. Fill in the last column of the table above using the percentages in the third col-umn. Use the growth rate in one year to calculate the estimated population in the nextyear.

(d) How much of a difference is there between the values in 2010 of these two functions?(e) We might wonder whether this difference might be significant enough to affect decisions

that governments might make concerning the availability of food, energy, and otherresources, etc. Make a rough estimate of how much “the world” would save in food costs

4. Functions 95

if the population in 2010 is the lower of the two estimates instead of the higher. Makethe most reasonable assumptions you can in developing your estimate, and explicitlysay what these assumptions are and how you arrived at them.

6. Composition of Functions. In this activity, you will gain practice in working with com-position of functions.

(a) Let f(x) = 2x + 1.i. Find (f ◦ f)(x), (f ◦ f ◦ f)(x), and (f ◦ f ◦ f ◦ f)(x). Do you notice any pattern?ii. In order not to have to write lots of f ′s and ◦’s, let’s define f (n)(x) to denote the

composition of f with itself n times. So, for example, f (3)(x) = (f ◦ f ◦ f)(x).Continue with the work you started in part (a) i., computing formulas for f (n)(x)for larger values of n until you see a general pattern. Describe this pattern as bestyou can in words or symbols.

iii. If you did not already do so in part ii., write a general formula for f (n)(x). Thisformula can include the variable x and the letters f and n. For example, youranswer might be something like f (n)(x) = (n− 1)x + 2n.

(b) Repeat part (a) using the function g(x) = 3x + 2.

7. Weight versus Height.For the first two years of a child’s life, the function y = w(h) =1.4h−20 can be used to approximate the average weight y, in pounds, of a child whose heighth (or length) is given in inches.

(a) Find the value for w(25) and describe what it means in terms of height and weight of achild.

(b) Find the estimated weight of a child who is 30 inches in length.(c) Find the estimated height h of a child who weighs 30 pounds.(d) Suppose we wanted to create a new function which calculated the weight in kilograms

instead of pounds as the output, and still let the input height be measured in inches.i. There are approximately 0.45 kilograms in 1 pound. What is the average weight,

in kilograms, of a child that is 20 inches tall?ii. The new function could be described as y = 0.45w(h). Find an equation for this

new function, which gives the output y as a function of h.iii. What is the average height, in inches, of a child who weighs 12 kilograms?

8. Do people who read live longer? Based on data collected for countries in North andSouth America, the function y = S(r) = −.142r + 98.3 approximates the percent of femaleswho survive to age 10 in a given country S(r) as a function of the percent of women in thegiven country who do not know how to read r.

(a) Find the value for S(10) and describe what it means in terms of illiteracy and survival.(b) Find the estimated survival to age 10 for a country where the illiteracy rate is 2%.(c) Find the estimated illiteracy rate for women in a country where 90% of females survive

to age 10.(d) What is the estimated survival rate for a country where none of the women know how

to read?

5. Linear Functions

One should not increase, beyond what is necessary, the number of entities required toexplain anything.— Willam of Occam (Ockham)

A problem should be stated in its basic and simplest terms. In science, the simplesttheory that fits the facts of a problem is the one that should be selected.— Dr. James Schombert

We have looked at various ways to present data as well as various ways to represent functions.We will now look at different types of functions, what we will call “families” of functions. All thefunctions in a given family will share some general characteristics that define that family, muchlike all the animals in a given species share the characteristics of that species.

We will begin with the family of linear functions, the simplest and most common type offunction. Common examples of linear functions include the conversion of feet to meters, dollars toeuros, and temperature from the Fahrenheit scale to the Celsius scale.

Linear functions are important partly because they are the simplest types of functions to workwith, and partly because linear functions supply good models for many, many different types ofreal-world situations. When looking for a mathematical model, look linear first. In this chapter,we will explore some of the important characteristics of linear functions and how to determine if agiven function is linear, as well as some examples of linear models.

Definitions

What is a linear function? What are the characteristics that define the family of linear func-tions? The easiest answer is that linear functions have graphical representations which are straightlines. However, we want to be able to tell if a particular function is linear whether it is representedgraphically, symbolically, numerically, or even verbally.

Numerical and Graphical Representations of Linear Functions

Consider the the graph in Figure 1 of a linear function.Two points on the graph are labelled. We can see that for each one unit increase in x, the

y-value increase by 3. If the graph is linear, this property must hold for every 1 unit increasein x, otherwise the angle or steepness of the line would change. This leads us to the numericalrepresentation given in Table 1.

Table 1: Tabular representation of a linear function.

Input 0 1 2 3 4Output 2 5 8 11 14

The table reflects that for each 1 unit increase in x, the y-value increase by 3. What if thex-value increases by 2 units? In this case, the y-value increases by 6 units. Doubling the changein x also doubled the change in y.

In general, if we pick any two points either on the graph or in the table, the change in outputsdivided by the change in inputs will always be the same, namely 3. This happens regardless ofwhether the input is changing by 1 unit, by 4 units, or by any other number of units. For thisfunction, the change in output is always three times the change in input. We say that 3 is the rateof change of y versus x. One way to define linear functions is that they are continuous functions

96

5. Linear Functions 97

?x ?

-2

-1

0

1

2

3

4

5

6

7

x -3 -2 -1 0 1 2 3 4

1

3

Figure 1: Graph of a linear function that has a starting point of (0, 2) and whose change in outputis 3 times the change in input.

with a constant rate of change. Numerically, a linear function is a function where the ratio,change in outputchange in input , is constant over the entire domain of the function.

The symbol ∆ (the upper-case Greek letter delta) is commonly used to denote change in avariable. We can therefore refer to the change in x as ∆x (or “delta x”)and the change in y as ∆y(or “delta y”). Then,

∆y

∆x=

change in outputchange in input

denotes the rate of change for any linear function.This constant rate of change can be used to create a graphical representation of a linear function.

One way to specify a linear function (or, equivalently, to draw a straight line), is to pick a startingpoint (somewhere to put your pencil) and a rate of change (how much the output changes whenthe input changes by 1 unit).

For example, suppose you are asked to draw a line starting on the y-axis at the point (0, 1)with ∆y

∆x = 2, or in other words, the change in y will be 2 times the change in x. You just needto plot the starting point and a second point and connect them to draw your line. You are giventhe starting point, (0, 1), so you need to locate a second point. Since the change in output is 2times the change in input, you can begin at (0, 1), move to the right one unit and then move up 2times one unit or two units to locate the second point. This produces (0+1, 1+2) = (1, 3) as yoursecond point. Connecting these two points with a line gives you the graph shown in Figure 2.

Note that these two pieces of information, where to start and the constant rate of change, aresufficient for drawing the graph of our linear function. This is what makes linear functions simple.You only need two pieces of information to completely define the function.

We define these two pieces of information in general as follows. The y-intercept is the pointwhere the line crosses the y-axis (or, equivalently, the output when the input is zero). This can bethought of as our starting point. The slope is the constant rate of change or

slope =change in outputchange in input

=∆y

∆x=

y2 − y1

x2 − x1

where (x1, y1) and (x2, y2) are any two points on the line. For the linear function given in Table 1,the y-intercept is 2 (since this is the input when x = 0) and the slope is 3. For the linear functiongiven in Figure 2, the y-intercept is 1 and the slope is 2.

98

?x ?

-2

-1

0

1

2

3

4

x -3 -2 -1 0 1 2 3

1

2

Figure 2: Graph of a linear function that has a starting point of (0, 1) and whose change in outputis 2 times the change in input.

Linear functions can have a negative slope. For example, a slope of negative 3 means that, forevery increase of one unit in the input, the output decreases by 3 units. To draw a graph withslope −3 and y-intercept 2, we start at (0, 2), and for each one unit we go to the right (increasethe input by one), we go down three units (decrease the output by three). So the second point is(0+1, 2− 3) = (1,−1). Connecting these two points creates the graph of this linear function. (SeeFigure 3.)

?x ??+

-2

-1

0

1

2

3

4

x -3 -2 -1 0 1 2 3

1

-3

Figure 3: Graph of a linear function with a y-intercept of 2 and a slope of −3.

Example 1. Consider the functions represented in Table 2(a) and Table 2(b). One is a linearfunction and one is not. Can you tell which is which?

Solution: To determine if a function given numerically is linear, we need to determine if thereis a constant slope or rate of change, ∆y/∆x. The function given by Table 2(a) has a constantrate of change (or slope) of 2 over the entire domain, while the function given by Table 2(b) doesnot. In Table 2(a), as x increases by 1, y always increases by 2, so the rate of change is 2. Notethat Table 2(b) does follow a discernible pattern. This pattern is not linear, however.


Table 2: Deciding if a table of numbers represents a linear function.

x −2 −1 0 1 2y −5 −3 -1 1 3

∆y/∆x 2/1 2/1 2/1 2/1(a)

x -2 -1 0 1 2y 8 2 0 2 8

∆y/∆x −6/1 −2/1 2/1 6/1(b)

Symbolic Representations of Linear Functions

In this section, we will see that linear functions have symbolic representations of the formy = mx + b or equivalently f(x) = mx + b where m will equal the slope and (0, b) will be they-intercept. Let’s see why this is.

Consider the linear function from Table 1 above, given again here.

Table 3: Table 1 reproduced.

Input 0 1 2 3 4Output 2 5 8 11 14

The slope is 3 and the y-intercept is 2. Is f(x) = 3x+2 a symbolic representation of this function?

By definition, the y-intercept of the function f is f(0), which in this case equals 3 · 0 + 2 = 2.This is the same value we see in the table. By hand, we can also check that f(1) = 3 · 1 + 2 = 5,f(2) = 3 · 2 + 2 = 8, f(3) = 3 · 3 + 2 = 11, and f(4) = 3 · 4 + 2 = 14. So, this function f doescorrespond to the table, at least for the values we have.

We noted previously that a linear function in numerical form will always give the same changein y for each 1 unit change in x. We can add additional entries to the table using this property, andthen check that the symbolic form gives the same y values. Instead, let’s check that the symbolicform always produces a ∆y

∆x equal to 3.Let’s designate an arbitrary input x0. The output under the function f would be f(x0) = 3x0+2.

If we increase the input by 1, we get x0 +1 and a corresponding output of f(x0 +1) = 3(x0 +1)+2.Calculating the change in y divided by the change in x, we have

∆y

∆x=

f(x0 + 1)− f(x0)x0 + 1− x0

=3(x0 + 1) + 2− (3x0 + 2)

1= 3x0 + 3 + 2− 3x0 − 2 = 3

So the formula f(x) = 3x + 2 always results in an increase in y of 3 whenever x is increased by 1unit. Since it has the same slope of 3 and y-intercept of 2 as the function given by Figure 1 andTable 1, it must be the symbolic representation of that function.

We can use this same logic to show that the general form f(x) = mx + b will give a linearfunction with slope m and y-intercept b. If x = 0, we have f(0) = m · 0+ b = b. So, the y-interceptof f(x) = mx + b is b.

For the slope, we have

∆y

∆x=

f(x0 + 1)− f(x0)x0 + 1− x0

=m(x0 + 1) + b− (mx0 + b)

1= mx0 + m + b−mx0 − b = m.

100

This shows the slope of the function f is constant and equal to m.The conclusion is that the function f(x) = mx + b gives us the two pieces of information we

need, the slope and the y-intercept, for the linear function with slope m and y-intercept b.

Finding the equation of a line through any two points

Because the function f(x) = mx + b is linear with slope m and y-intercept b, we can find thelinear function in symbolic form given any two points. If we let (x1, y1) and (x2, y2) denote anytwo points, we can express the slope as

∆y

∆x=

change in y

change in x=

y2 − y1

x2 − x1

Using this fact, we can use the following procedure to find the equation.

• Find the slope using ∆y∆x = y2−y1

x2−x1

• Substitute the slope and the x and y values from either known point into the equationy = mx + b

• Solve for b

• Write the equation using the calculated m and b.

Example 2.Find the equation of the line through the points (2, 9) and (6, 1).

Solution:Following our procedure, we first find the slope. We have

∆y

∆x=

y2 − y1

x2 − x1=

1− 96− 2

= −2

so the slope is −2.Using the point (6, 1) and the slope of −2, we substitute into y = mx + b to get

1 = −2 · 6 + b

1 = −12 + b

1 + 12 = 13 = b.

So, b = 13. This gives us the equation y = −2x + 13, or in function notation, f(x) = −2x + 13.We can double-check that this is correct by checking that the two points we started with are validinput-output pairs. We have f(6) = −2 · 6 + 13 = −12 + 13 = 1. Also, f(2) = −2 · 2 + 13 = 9. So,this function does give us a linear function through both given points.

Verbal Representations of Linear Functions

Linear functions provide good models in a wide variety of contexts. Principally, any two vari-ables that show an approximately constant rate of change can be described by a linear function.In some cases, the rate of change might be precisely or nearly precisely constant.

For example, if you travel at a constant velocity of 75 mph, then the distance, d (in miles), thatyou travel is represented by the linear function

d = 75t,


where t is time (in hours). When we have driven for 0 hours (t = 0), we have gone 0 miles (d = 0).So the y-intercept is zero. The slope is 75 since the constant rate of change (i.e. velocity) is 75mph. For every 1 hour change in the input (for every hour you drive), your output (distance)increases by 75 miles.

In a situation where two variables do show a constant rate (or approximately constant) ofchange, we can find the equation by determining two pieces of information, the slope and the y-intercept. In a physical (or verbal) situation, the y-intercept will be your starting point (i.e. youroutput when your input is zero) and the slope will be your constant rate of change.

Example 3. Suppose you call a professional to take care of a furnace problem. He charges $20per visit plus $30 per hour. Let C be the function whose input is n, the number of hours worked,and whose output is C(n), the total cost.

1. Explain why this is a linear function.

2. Find an equation for C.

Solution:

1. This is a linear function since the rate of change (in this case, the charge per hour) is constant.

2. To find the equation, we need to find the slope and the y-intercept. The slope, or rate ofchange, is the $30 charge per hour. The y-intercept is the cost for the furnace person to showup or the starting cost which is $20. So the equation for this function is

C(n) = 30n + 20

where C(n) is the total cost (in dollars) and n is the number of hours the plumber works.

Note in the previous example that the units for the slope is dollars per hour, and that the unitsfor the output is dollars, and the units for the input is hours. This general pattern works for alllinear modeling functions. We always have:

The units for the slope m are unit of output per unit of input.

Going back to our distance function d = 75t, the units for the output d were miles and the unitsfor the input t were hours. The units for the slope is miles per hour.

Example 4. Suppose you are currently spending about $80 per month on electricity, and thatover the past 12 months, your cost has been increasing about $1.20 per month. What linearfunction would model your electricity costs?

Solution: If we let x represent months with x = 0 the current month, then $80 represents youry-intercept or starting point. The $1.20 represents your constant rate of change. Assuming thisrate remains constant at about this level into the future, your electric costs y as a function of x willbe y = 1.20x + 80. So for example, 20 months from now your electric bill would be approximately1.20 · 20 + 80 = $104.

Linear functions are often the basis for converting units of measurement. This is because therate of change for units of measurement is nearly always constant. For example, there are always12 inches per foot, 5280 feet per mile, or 4 quarts in a gallon. So, to convert from feet to inches,we multiply the number of feet by 12 to get the number of inches. Symbolically, if i is the numberof inches and f is the number of feet, we have i = 12f . This is a linear equation with slope 12 andy-intercept 0. If we put in the units, we have

i inches =12 inches

1 foot× f feet.

102

We can find similar conversion equations for feet to miles (slope is 15280

milesfoot ), or gallons to quarts

(slope is 41

quartsgallon ).

How long is a billion seconds in years? (Go ahead and take a guess.)

Let’s look at linear functions used for converting one unit of time into another. To start with,let’s convert seconds to years.

There are 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, and 365 days in ayear. So,

number of seconds × 1 minute60 seconds

× 1 hour60 mins

× 1 day24 hours

× 1 year365 days

= number of years

or (multiplying all the fractions together)1

31, 536, 000s = y

where s represents number of seconds and y number of years. Notice, again, the cancellation ofunits to produce the correct unit for the output.61

If s = 1, 000, 000, 000 = 109, then

y =1

31, 536, 000109 ≈ 31.71 years

or 31 years and about 8 and 1/2 months.62 Also, we should mention that Appendix 4 contains alist of some other “large numbers” with comparisons for you to get a sense of “how large is large?”

Also notice that the y-intercept is again zero. This is because zero in the first unit of measure-ment corresponds to zero in the second unit of measurement. Zero seconds means zero years. Zerofeet means zero inches. Will the y-intercept for a linear conversion function always be zero?

The answer is “no.” One example is the conversion from temperature in degrees Celsius totemperature in degrees Fahrenheit. The Celsius scale was created so that 0 degrees Celsius is thefreezing point of water and 100 degrees Celsius is the boiling point of water. So, 0◦ C equals 32◦F, and 100◦ C equals 212◦ F. This means that the linear function that converts from temperaturein the Celsius scale to the Fahrenheit scale must contain the points (0, 32) and (100, 212). They-intercept will be 32. We find that the slope is 212−32

100−0 = 1.8. So the conversion equation isthe linear equation F = 1.8C + 32. The slope of 1.8 means that a change of 1 degree Celsius isequivalent to a change of 1.8 degrees Fahrenheit.

Solving Literal Equations

In Chapter Four, we discussed how to solve equations like 2x + 5 = 21. This equation had onevariable, and we were able to find the single value for x which solved the equation.

Many times, we will consider equations that have more than one variable or “letter,” likey = mx + b. When an equation has more than one variable or undetermined parameter, we callit a literal equation. Literal equations are solved using the same methods as single-variableequations. One difference is that the solution will not be a number but an expression. Anotherdifference is that one must specify which of the several variables one is going to solve for.

61Using cancellation of units is a good way of deciding if you want 1 minutes60 seconds

or 60 seconds1 minute .

62If you are 20 now and want to be a billionaire by the time you are 50, you need to earn more than $1 per second,on average, for 30 years.


Example 5. Solve the following literal equations.

1. Solve y = mx + b for the variable m.

2. Solve B = P (1 + r)t for P .

Solution:

1. To solve for m we can add, subtract, multiply or divide by equals on both sides, as long aswe don’t multiply or divide by 0. It often helps to think of using the reverse of the usualorder of operations. To start, we will subtract b from both sides.

y = mx + b

y − b = mx + b− b = mx

Then, we divide by x.

y − b

x=

mx

x= m

our solution is that m = (y − b)/x.

2. The equation B = P (1 + r)t looks a little more complicated than the first, but is actuallyeasier to solve. Simply divide by (1 + r)t.

B = P (1 + r)t

B

(1 + r)t=

P (1 + r)t

(1 + r)t= P

So, P = B(1+r)t .

Variations in symbolic form

Sometimes a function that actually is linear may be “in disguise”, that is, written symbolicallyin a format other than y = mx + b. Given any literal equation with variables x and y, if we canrewrite it in the form y = mx + b, then we know we have a linear function.

One way to do this is to algebraically solve for y (or whatever letter represents the output)before deciding if your function is linear. Consider the equation

y − 5x

= 3.

This does not “look” like the equation of a line, but if we solve for y (by multiplying both sides byx and then adding 5), we get

y = 3x + 5

which is the equation for a linear function with slope 3 and y-intercept 5.63

63The original equation y−5x

= 3, excludes the input x = 0 because of division by zero. So, technically, this is alinear function which is not defined for x = 0.

104

What about the equationy − 5

x= 3 + x?

To solve for y we multiply both sides by x and then add 5. This gives us the equation

y = 3x + x2 + 5.

This is not a linear function since the equation has an x2 term. To be safe, always solve theequation for y (or whatever letter represents the output) and see if it looks like y = mx + b beforedeciding if the function is linear.

Regression and Correlation

When we looked at scatterplots in Chapter Two, we introduced the idea of strong and weakassociations. In Chapter Four, we saw a couple of examples of scatterplots where we included whatwe called “the linear function that best fits the data.” In this section, we will make both of theseconcepts more precise by quantifying what we mean by “best fit” and “strength of association.”

The scatterplot in Figure 4 shows how U.S. energy consumption has been increasing since 1982.Although the data is not precisely linear, it is approximately linear and so we can fit a line to thisdata by drawing a line that is close to all the points. We can then use the equation of this line asa symbolic description of the data. The line plotted in the graph is called the linear regressionline (also know as the least-squares line) for this data. The equation is E(t) = 0.19t + 10.6,where t represents years from 1980 and E(t) represents total energy consumption in quadrillionsof BTU’s.64 Let’s look at where this equation came from.Total U.S. Fossil Fuel Consumption 1982 to 2002Billions of Barrels of Oil Equivalent

101112131415161980 1985 1990 1995 2000 2005

Figure 4: U.S. Energy Consumption

The Method of Least Squares

The most common method of determining the equation of the line that best fits a set of datais called the method of least-squares. The resulting line is referred to as the least-squaresregression line or simply the (linear) regression line. Formally, a least-squares regression line is

64BTU is short for British Thermal Unit. Eight gallons of gasoline contains the equivalent of about 1 millionBTU’s of energy.


a line that best fits two-variable data by minimizing the sum of the squares of the vertical distancesbetween the points in the scatterplot and the line.

To illustrate this, let’s consider the data shown in Table 4, which gives heights and arm spansof a small sample of college students.

Table 4: Heights and arm spans (in centimeters) of sample college students.

Height 152 160 165 168 173 173 180 183Actual Arm Span 159 160 163 164 170 176 175 188

A graph of the data from Table 4 is shown in Figure 5.65

Arm

Spa

n

155

160

165

170

175

180

185

190

Height150 155 160 165 170 175 180 185

ArmSpan = 0.926Height + 18.3; Sum of squares = 382.6


Figure 5: Scatterplot and sample line for student arm span versus height data

We have also fitted a line to this data, but this is not the best fit line. Notice, for example, thatwhile the line goes through two of the points, all the rest of the points are below the line. Finally,we have also included the “squared errors” in the graph. For each point, a vertical line from thatpoint is drawn to the graph; the length of this vertical line is called the error for this data point.Then, a square is created using this line as one of the sides. The closer a point is to the line, thesmaller the error is, and so the smaller the square. Adding up the area of all these squares we getwhat is called the sum of squared errors, or just sum of squares for short. For our line, the sumof squared errors is equal 382.6. This number is a measure of the overall error between the lineand the data points.

The regression line is the line which makes this sum of squares which measures the overall erroras small as it can possibly be. Now, of all the infinitely many lines that we could draw, how canwe find this best fit line?

It is possible to find this line “by hand,” using ideas from calculus. However, we will usetechnology which is programmed to do linear regression. For example, you can find linear regression

65This graph was created using a statistical software package called Fathom

106

lines using most graphing calculators, the Excel spreadsheet program, or most any statisticalsoftware package. We found that the least square regression line for the arm span data is

arm span = 0.875× height + 21.4.

Figure 6 shows the data again, but now with the linear regression line, instead of the line weused above. Notice that the sum of squares is now only 125.3.

Arm

Spa

n

155

160

165

170

175

180

185

190

Height150 155 160 165 170 175 180 185

ArmSpan = 0.875Height + 21.4; r 2̂ = 0.82; Sum of squares = 125.3


Figure 6: Scatterplot with Regression Line

Table 5 is the same as Table 4, except that it gives the values predicted by the linear regressionline, and the errors and squared errors of our line, i.e. the difference between the actual output(using the data points) and the predicted output (using the equation of the line). The sum ofthe squares of all these errors is 125.42, essentially the same as the 125.3 given in the graph (thediscrepancy is due to round off error).

Table 5: Heights and arm spans with regression predictions and errors.

Height 152 160 165 168 173 173 180 183Actual Arm Span 159 160 163 164 170 176 175 188

Predicted Arm Span 154.4 161.4 165.8 168.4 172.8 172.8 178.9 181.5using the line

Error 4.6 -1.4 -2.8 -4.4 -2.8 3.2 -3.9 6.5Squares 21.16 1.96 7.70 19.36 7.70 10.40 15.21 41.93

One more note on the regression line. Notice that the slope of the line is 0.875. Recalling thatthe slope is the rate of change between the two variables, this means that the linear regressionline predicts that each one centimeter increase in height results in an increase in arm span of .875centimeters. Of course, since the line does not follow the data exactly, this is only an approximaterate of change, but we can consider it to be the best estimate of the rate of change between thesetwo variables.


Correlation

The two examples of regression lines that we have looked at so far do a reasonably good job ofpredicting values for inputs close to our data points. Both lines visually fit the data reasonably well,and give predictions reasonably close to the actual data values. However, data cannot always beexpected to behave so nicely, and regression lines may or may not fit the data well. For example,in Chapter Two, we considered a scatterplot showing malnutrition versus poverty, and we havereproduced it here in Figure 7 along with the linear regression line.


0

5

10

15

20

0 10 20 30 40 50 60 70 80

Figure 7: Percent of Severely Underweight Children Versus National Poverty Rate.

As noted in Chapter Four, there does seem to be a general increasing trend, but the associationis certainly not as strong as in our previous two examples. We can see that the regression equationwill not give very accurate predictions since many of the data points are not very close to the line.

The standard measure of how well a regression line fits a set of data is called the correlationcoefficient, or correlation for short. This coefficient measures both the strength and direction ofa linear relationship between two numerical variables. The height and arm span data in Figure6 is said to be highly correlated while the malnutrition and poverty data in Figure 7 has a lowercorrelation.

Correlation is usually denoted by the letter r. Correlation is always a number between −1 and1. If the correlation is close to −1 or 1, then the data points are highly correlated, i.e. the plotteddata points lie very close to a line indicating a very strong association. In fact, if r = 1 or r = −1,the data all lie exactly on a straight line and the regression line perfectly predicts the outputs.If the correlation is close to 0, then the data points have little correlation, i.e. the plotted datapoints do not follow a linear pattern and there is no association or a very weak association. If ris sufficiently larger than 0, then there is a positive association between the two variables and theregression line will have a positive slope. If the correlation is sufficiently negative, then there is anegative association between the two variables and the regression line will have a negative slope.

The correlation of the height and arm span data shown in Table 5 and Figure 6 is r = 0.904while the correlation of malnutrition and poverty data is r = .369.66 This means that the heightand arm span data is highly correlated and has a positive association, i.e. people who are tallerhave longer arm spans. The malnutrition data has a much lower correlation, but there is still apositive association.

You can often compare correlations simply by looking at scatterplots of the data. Figure 8shows a number of different scatterplots with corresponding correlations.

66You might notice in Figure 6, it is noted that r2 = .82. Sometimes r2 is used instead of r because it can beinterpreted to mean “the percentage of the variation in the dependent variable which is explained by the independentvariable.”

108

Y1 = -1.26X + 19.8; r2 = 0.99

0

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0 2 4 6 8 10 12 14 16 18 20X


Y2 = -0.886X + 17.2; r2 = 0.46

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Y3 = -0.0980X + 12.4; r2 = 0.011

0

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r = -.99 r = -.68 r = -.11

Y4 = 0.730X + 11.4; r2 = 0.25

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0 2 4 6 8 10 12 14X


Y5 = 1.45X + 18; r2 = 0.64

10

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0 2 4 6 8 10 12 14 16X


Y6 = 0.786X + 15; r2 = 0.90

0

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0 2 4 6 8 10 12 14 16 18X


r = .5 r = .8 r = .95

Figure 8: Examples of various scatterplots and their corresponding correlation.

As with the least-squares regression equation, there are rather complicated formulas to deter-mine the correlation.67 However, as with regression lines, we will rely on graphing calculators orsoftware packages such as Excel to compute these numbers.

Correlation is often confused with causation. An example of two variables that have a causalrelationship would be Amount Earned versus Hours Worked. An increase in the number of HoursWorked will in general cause an increase in the Amount Earned. If you measured these variablesin a particular month for a number of people, the results would vary, but you would have a definitestrong positive correlation, and this correlation is the result of the causal relationship between thevariables.

However, two variables can be highly correlated without there being any causal relationship. Forexample, if you were studying the verbal ability of elementary school students, you would probablyfind a high positive correlation between a child’s height and his or her vocabulary, measured inwords. This does not mean that the growth of a child causes the child to learn more words. Thecorrelation is a result of the child growing and learning more words as they age. Correlation ismerely a numerical measure of the relationship between the data.

To establish causation, one would need to provide proof, or at least a plausible explanation,why change in one variable causes a corresponding change in the other variable. For example, anincrease in vocabulary probably has a causal relationship with standardized verbal test scores forchildren.

Correlation and changing the data set

In Chapter Two, we also commented that one might divide the points in our malnutrition versuspoverty scatterplot into two groups, one consisting of countries with a poverty less than 25% andthe second containing all the other countries. We noted that, looking at only the second group,there seems to be almost no pattern to the data. We can quantify this using correlation. If we look

67The formula for correlation involves the sum of squared errors, as well as the means and standard deviations ofthe two variables.


at only the data points in the second group, the correlation is r = 0.045, instead of the r = 0.369we obtained using all the data points, showing there is almost no association within the secondgroup of higher poverty countries.68

By excluding some data points, we have drastically altered the correlation and so possibly theconclusions we might reach based on the data. This is an important point to keep in mind. Whenconsidering correlation, it is a good idea to consider whether the selection of which data points toinclude has been biased, either intentionally or unintentionally.

One further point with respect to correlation is that it is dependent not only on the strengthof the association between the two variables, but also on the number of data points. To give anextreme example, if you do linear regression on a data set with only two points, you will always69

get a correlation of 1 or -1. This does not mean that there is really any relationship between thetwo variables. It just means that, given any two points, there is a line that will go exactly throughthose points. If we had picked any two points out of our malnutrition and poverty data, we wouldhave gotten a correlation of either 1 or -1.

The tendency will be that the more data points you have, the lower the r-value will be. Thismeans that the larger the data set, the smaller the r-value needs to be in order to indicate a strongcorrelation. Table 6 shows a few of what are called critical values of r based on the number of datapoints n.70 For a data set with n points, an r-value above the critical value for that n indicates thatyou can be confident there is a definite positive association. For now, it is enough to be aware thata large data set could have a stronger association than a smaller data set with a higher correlation.For example, Table 6 indicates that a data set with 10 data points and r = 0.632 would have thesame strength of association as a data set with 47 points and r = 0.288.

n r n r n r

3 .997 13 .553 27 .3815 .878 15 .514 37 .32510 .632 20 .444 47 .288

Table 6: Critical Values for r

As a visual comparison with the previous graphs, Figure 9 shows graphs with 47 points, insteadof the 13 points in the graphs in Figure 8.

Y1 = 0.796X + 6.4; r2 = 0.90

0

10

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30

40

50

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0 10 20 30 40 50X


Y2 = 0.646X + 14; r2 = 0.64

0

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Y3 = 0.457X + 15; r2 = 0.25

0

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r = .95 r = .80 r = .5

Figure 9: Examples of various scatterplots with their corresponding correlation coefficients.

68Again, we are just using Excel to compute this.69unless the two points have the same x-coordinates or the same y-coordinates70Values are from a larger table found in Gordon, et. al.

110

Application: Carbon Emissions from Fossil Energy Consumption

Table 7 gives carbon dioxide emissions from energy consumption for the commercial sector inthe United States.71 There are two variables, one for non-electric energy usage, and one for totalenergy usage from all sources. Non-electric includes direct use of petroleum products, coal, andother fuels besides electricity.

Table 7: Carbon emissions from the U.S. commercial sector from 1990 to 2003.

Year 1990 1995 1996 1997 1998 1999 2000 2001 2002 2003Non-electric 224 228 237 237 220 222 234 228 226 231

Total 777 837 869 912 930 944 1,004 1,021 1,020 1,026

A scatterplot Total Emissions versus Year is shown in Figure 10. We measured time as yearssince 1990. So 1990 is represented by 0, 1995 is represented by 5, etc. These points seem toportray a linear relationship. The least squares regression line for this data is y = 21.9x + 757,with correlation r = 0.975.

Total Carbon Dioxide Emissions Millions of Metric Tons

0

200

400

600

800

1000

1200

0 2 4 6 8 10 12 14

Figure 10: Total Carbon Dioxide Emissions for the U.S. Commercial Sector

Example 6. What is the physical interpretation of the slope and y-intercept of the least squaresregression line?

Solution: The y-intercept is 757, which means that at time 0 (the year 1990), approximately257 million metric tons of carbon dioxide were emitted from energy consumption by the U.S.commercial sector. This is a little below but reasonably close to the actual number million metrictons emitted in 1990 given in Table 7. The line has a slope of 21.9 which means the equationpredicts the amount of emissions increases an additional 21.9 million metric tons each year.

Looking at this trend, it is easy to see why environmental groups are becoming increasinglyconcerned with the need for the government to set higher standards for the commercial sector.If this pattern continues, carbon emissions just from the commercial sector will be 1414 millionmetric tons in the year 2020, almost double what they were in 1990.

71Data from the Energy Information Administration

Reading Questions for Linear Functions

1. What is the definition of a linear function?

2. What 2 pieces of information determine a line?

3. The graph below gives the y-intercept, rise, and run for a line. Find the equation of this line.

x ?

-2

0

2

4

6

8

x

-3 -2 -1 0 1 2 3

(0, 3)

2

3

4. Suppose y = f(x) is a linear function whose slope is 2. If f(1) = 5, what is f(2)? f(3)?f(10)?

5. Let y = f(x) be a linear function whose y-intercept is −1 and whose output changes by 3when the input changes by 1. What is f(2)?

6. Suppose f(x) = 5x + 4.

(a) How do you know f is a linear function?(b) What is the slope?(c) What is the y-intercept?(d) What is f(0)?(e) What is f(1)?

7. Sketch each of the following. Label at least 3 points on your graph.

(a) A linear function with y-intercept 0 and slope 13 .

(b) A linear function with y-intercept 1 and slope −2.

8. Let A,B, and C be three points such that the slope between A and B equals the slope betweenB and C. What, if anything, do you know about the slope between A and C? Explain.

9. Find the equation of the line through the points (5, 3) and (13, 9).

10. Determine if each of the following tables represent a linear function. If so, find an equationfor the line. If not, explain why.

(a)Input −2 −1 0 1 2

Output 3 0 −3 −6 −9

111

112

(b)Input −2 −1 1 2

Output −4 0 8 12

(c)Input −2 −1 0 1 2

Output 12 10 8 10 12

11. Find the equation of the line for each of the following.

(a) The y-intercept is 5 and the slope is −14 .

(b) It contains the two points (0, 10) and (1, 11).

(c)Input −2 −1 0 1 2

Output 0 2 4 6 8

(d)

x ? +

-1

0

1

2

3

4

5

6

7

8

x -8 -6 -4 -2 0 2 4 6

12. Determine whether or not each of the following represents a linear function.

(a)y − 1x + 2

= 4

(b) y − 4 = 1.2(x + 3)

(c)x + 3

y= 16

13. A calling card offers a price of $0.23 cents per minute at any time during the day with amonthly service charge of $1.00.

(a) Write a linear equation (including the $1.00 monthly service charge) where the input isminutes and the output is total monthly cost.

(b) Describe the practical interpretation of the y-intercept.(c) Using your equation from part(a), calculate the total cost for a month if you talk, on

average, for 20 minutes a day. Assume the month is thirty days long.

14. Give the linear equation that will convert from the first unit to the second.

(a) centimeters to inches (2.54 centimeters ≈ 1 inch)(b) yard to inches (1 yards = 36 inches)(c) degrees Fahrenheit to degrees Celsius. [Hint: Use the points (32, 0) and (212,100).]

15. We are interested in converting from miles per hour to feet per second.

(a) Convert 70 mph to feet per second.


(b) Write a linear conversion function whose input is the velocity in miles per hour andwhose output is the velocity in feet per second.

16. Suppose the linear function y = 2, 500x − 15, 000 represents the average salary y of a U.S.citizen who has complete x years of school. Explain carefully, using the specific parametersgiven in the equation and appropriate units, what the slope and y-intercept of this equationmean.

17. Suppose a baby boy’s birth weight was 8.25 pounds and he gained 2.1 pounds each monthfor the first few months of his life. The function describing his weight is linear and could bewritten as

y = 2.1x + 8.25,

where x is his age in months and y is his weight in pounds.

(a) Give the slope for the linear equation, and explain what the slope means in terms ofweight and age.

(b) Determine the baby’s weight after 3 months.(c) Determine the child’s weight after 36 months. Is your answer reasonable? Explain.(d) If the baby’s birth weight was 8.5 pounds instead of 8.25, how would the function change?(e) If the slope of the function were 2.0, what would this mean?

18. The following is the graph of a scatterplot and its regression line. The data is from studentsin two algebra sections in the fall of 2007. The relationship is scrabble points versus numberof letters in the last name for each student.

0 2 4 6 8

101214161820222426

0 2 4 6 8 10 12

Scatter Plot

Ltrs_Last

(a) One student claimed that the regression equation gave a value of 3 scrabble points fora student with 2 letters in their last name. Is this correct? Why or why not?

(b) Another student claimed that the regression equation gave a value of 15 scrabble pointsfor a student with 10 letters in their last name. Is this correct? Why or why not?

(c) Which of the following is the equation of the regression line for this data? For eachanswer you do not choose, explain why.

i. y = 4.2x + 1.1ii. y = 1.1x + 4.2iii. y = 10.1x + 7.5iv. y = −1.1x + 4.2

114

(d) Using your answer from the previous question, give the slope and y-intercept of theregression equation and explain what each of these mean in terms of scrabble points andnumber of letters in a person’s last name.

19. The table below gives the data for U.S. per capita energy consumption but in quadrillions ofBTU’s, rather than billions of barrels oil equivalent as in Figure 4 in the text.

Year 82 83 84 85 86 87 88Consumption 73.2 73.1 76.7 76.4 76.7 79.2 82.8



(a) Find the linear regression equation for this data, letting x = 0 represent the year 1980.(b) What is the correlation coefficient for this data?(c) Explain the meaning of the slope using appropriate units.(d) Explain the meaning of the y-intercept using appropriate units.

20. Can you ever have a least-square regression line where all the data points are below the line?Explain.

21. What does correlation measure?

22. Give an example of two variables not mentioned in the reading that have a negative correla-tion.

23. If a person’s arm span was always 3 inches shorter than his or her height, what would be thecorrelation between arm span and height? Explain.

24. Explain what is wrong with the following statements about correlation and regression.

(a) The regression equation that describes the relationship between the age of a used FordMustang and its value is y = −1000x + 10, 000, where the input is the age of theautomobile (in years) and the output is its value (in dollars). The correlation describingthis relationship is 0.83.

(b) The correlation between the number of years of education and yearly income is 1.23.

25. Just because two variables are highly correlated does not mean that one necessarily causesthe other. There may be something else that is causing one or both events.

(a) There is a high correlation between a child’s height and his or her ability in mathematics.Does this mean that a child’s physical growth is causing the increased ability in math?Explain.

(b) There is a high correlation between the outside temperature and the number of coldspeople get. Does this mean that the number of colds people get causes the temperatureto decrease? Explain.

(c) There is a high correlation between the number of drinks someone has at a bar andincidents of lung cancer. Does this mean that drinking causes lung cancer? Explain.


26. Match the following correlations with the corresponding scatterplot.

(i) r = 0.98 (ii) r = 0.8 (iii) r = 0.4 (iv) r = 0.1 (v) r = −0.5 (vi) r = −0.9

02468

101214161820

X0 2 4 6 8 10 12 14


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X0 2 4 6 8 10 12 14


02468

101214161820

X0 2 4 6 8 10 12 14


(a) (b) (c)

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X0 2 4 6 8 10 12 14


(d) (e) (f)

27. Input the following data into your calculator.

x 1 2 3 4 5 6 7 8y 5 7 9 11 13 15 17 19

(a) Determine the regression equation and correlation for the data.(b) What does the correlation tell you about the data and the regression equation?

28. In the reading, we found that the least squares regression line for the Carbon Emissions datashown in the following table was y = 21.9x + 757, where x was the number of years since1990 and y was carbon emissions in millions of metric tons for that year.

(a) The correlation for the regression line was r = 0.975. What information does this giveyou about the carbon emissions data?

(b) Use the regression line to estimate the level of carbon emissions in the commercial sectorin 2008. How confident are you about the accuracy of your answer? Explain.

29. The following table gives three variables for 26 countries in North and South America.72 Allthe variables measure the female population in the given country. The variables are:

• Illiteracy Rate, measured as the percent of females over 15 years of age who are func-tionally illiterate.

72Illiteracy data is from the World Bank. Life expectancy data is from the World Health Organization. All variablesare from circa 2000

116

• Additional life expectancy at age 10, which represents how many additional years afemale aged 10 can expect to live, on the average.

• Survival to 10, the percentage of newborn females expected to live to age 10.

Percent PercentIllit. Additional Surviving Illit. Additional Surviving

Country Rate Life Exp. to Age 10 Country Rate Life Exp. to Age 10Barbados 0.3 68.5 0.99 Panama 8.8 68.2 0.98Guyana 1.9 61.9 0.93 Jamaica 9.3 67.8 0.98Uruguay 2.0 69.1 0.98 Ecuador 10.1 66.9 0.96Trin. & Tob. 2.4 64.8 0.99 Mexico 10.9 68.3 0.97Argentina 3.2 69.5 0.98 Brazil 13.2 65.1 0.96Cuba 3.4 68.2 0.99 Peru 14.8 65.4 0.95Bahamas 3.7 65.9 0.98 Dom. Rep. 16.3 65.3 0.95Costa Rica 4.4 70.0 0.98 Bolivia 20.8 59.5 0.91Chile 4.4 70.3 0.99 El Salvador 24.0 66.0 0.96Belize 6.9 67.3 0.97 Honduras 25.0 64.4 0.95Paraguay 7.8 66.7 0.97 Nicaragua 33.3 64.3 0.96Venezuela 8.0 68.3 0.98 Guatemala 38.9 62.6 0.94Colombia 8.4 66.8 0.98 Haiti 52.2 52.6 0.89

(a) Consider the relationship Additional Life Expectancy Versus Illiteracy Rate. What is thecorrelation? What does this number tell you about the relationship between illiteracyand life expectancy?

(b) Consider the relationship Percent Surviving to Age 10 Versus Illiteracy Rate. Whatis the correlation? What does this number tell you about the relationship betweenilliteracy and life expectancy?

(c) How does the strength of the first relationship compare with the second? Is this sur-prising?

(d) Make separate scatterplots of both relationships. In each case, which countries seem tobe outliers? If these outliers were removed from the data, what would happen to thecorrelation coefficients? (You could do this by recalculating the coefficients, but youshould be able to answer correctly without doing this!)

(e) Give the slopes for the linear regression equations for both relationships and explainwhat these slopes mean in the given context.


Linear Functions: Activities and Class Exercises

1. Protein Requirements Revisited. In 1985 the World Health Organization (WHO) pub-lished revised daily protein requirements which translate into 56g of protein a day for a 75kgman, and 48 grams for a 64 kg woman

(a) Assume that the amount of protein a person requires is proportional to the person’sweight. Using this assumption and the information given above, find the linear functionexpressing the protein requirement for women as a function of weight.

(b) Explain what the slope of the line you found in part a) means. Do you get the sameslope whether you use the data given for women as for men?

(c) Calculate how many grams of protein each of the members of your team requires.(d) The table below gives the number of calories for each gram of protein for a variety of

plant foods and lean ground beef. 73

No. of cals Number of cals totalFood per gram protein for protein RDI

Spinach 5Broccoli 10Collards 12.5

Lettuce(1) 14Celery 20

Corn (3) 28Potato 44

Soybeans 12Lentils 13.5

Peas (4) 15Peanuts 21

Wheat(2) 25Oatmeal 24

Sunfl. sds. 27Almonds 28Grd. beef 11

(1) Iceberg variety. (2) Hard Spring Wheat. (3) fresh. (4) frozen

Given the information in the second column, one can calculate how many calories fromany particular food item one would need to eat to get their entire RDI of protein fromthat food item. Calculate this value for each of the food items in the table for a 64kgwoman, putting the results in the third column of the table.

(e) Which of the food items in the table could a typical 64kg women use to fulfill her proteinneeds without going over the calorie limit of 2100 recommended for such an individual?

(f) What additional important information about these food items would be helpful inplanning a daily menu which fulfills basic nutritional needs?

73Data from the USDA, was accessed at http://www.caloriecountercharts.com/chart3a.htm

118

2. World Population Growth Rates. As noted in the Activities for Chapter Four, theannual growth rate of world population has been decreasing for a number of years, and thegraph below shows the percent growth rate for world population, going back to 1950 andprojected out to 2050.


(a) After about 2000, and certainly after 2020, the graph is very nearly linear. Looking atjust the portion of the graph from 2020 on, estimate the slope of the graph. In words,describe what the slope is telling you about world population growth rates.

(b) Using the slope from part (a), and assuming the graph continues in the same way,estimate when the growth rate will reach 0%.

(c) What would it mean for the population growth rate to be negative? Do you think thisis possible? Why or why not?


3. Anscombe II. The graphs below are scatter plots of 4 different two-variable data setsproduced by Anscombe74 as examples related to linear regression. You may have done anactivity using these graphs in chapter three. We have labeled the 4 plots A, B, C, and D tothe lower left of each plot.

A.

456789

101112

B

4 6 8 10 12 14 16 18A


B.

3456789

10

C

4 6 8 10 12 14 16 18A


C.

56789

10111213

D

4 6 8 10 12 14 16A


D.

56789

10111213

F

8 10 12 14 16 18 20E


(a) For each of the 4 graphs, make your best prediction of the correlation coefficient r forthe data. Using your estimates, rank the 4 data sets from highest to lowest with respectto your estimated correlation coefficients.

(b) On each of the 4 graphs, draw in your best intuitive guess for the linear regression line.Then, for each of the 4 lines, estimate the slope of the line.

(c) In fact, all four data sets have exactly the same correlation coefficient of r = .82, andthey all have exactly the same linear regression line of y = .5x + 3.0. Compare yourestimates for r and for the slope of your estimated linear regression lines with the actualvalues.

(d) Even though the correlation coefficients and regression equations are identical, the linearregression model is not equally appropriate in all cases. For which of the 4 data sets isthe linear regression model a good model?

(e) Consider the C graph (with variable D on the vertical axis). If this graph was based onreal-world data, what explanation might there be for the outlier in this graph?

74Data from Anscombe, F.J., Graphs in Statistical Analysis, American Statistican, 27, 17-21

120

4. Food Insecurity. The USDA uses the following definition for food insecurity.

Food insecurity is limited or uncertain availability of nutritionally adequate andsafe foods or limited or uncertain ability to acquire acceptable foods in sociallyacceptable ways.

One way to measure this is using Percent Food Insecure Households (PFIH), which gives thepercent of all households which are experiencing food insecurity. In this activity, we’ll lookat PFIH and its relationship with four other variables. These variables, and scatter-plots ofPFIH as the dependent variable versus them are given below. Each point represents one ofthe 50 states. South Dakota’s PFIH was 8.07% in 2000. The other four variables are:

• Per Capita State Tax Revenue• Percent Drop in Welfare Rolls 1995-2002• Per Capita Personal Income• Child Poverty Rate

PFIH vs Per Cap Personal Income

0

2

4

6

8

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18

20,000 25,000 30,000 35,000 40,000 45,000

PFIH vs Child Poverty Rate

0

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18

5 10 15 20 25 30

PFIH vs Per Capita State Tax Revenue

0

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1,000 1,500 2,000 2,500 3,000 3,500

PFIH vs Drop In Welfare Rolls

0

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0 20 40 60 80 100


A few words about the variables.

• Per Capita State Tax Revenue is equal to the total amount of state tax revenues dividedby the population. For South Dakota, this was $1228 in 2000, lowest among the fiftystates.75

• Percentage Drop in Welfare Rolls measures the percent decrease in the number of fami-lies on welfare from 1995 to 2002. The reduction in welfare rolls in each state is largely aresult of the Welfare Reform legislation that was passed during the Clinton Administra-tion. South Dakota’s drop was 62%. On average, more than 6 out of every 10 familieson welfare in 1995 were dropped from welfare rolls. However, one must keep in mindthat this was not a fixed population, and some families went on welfare while othersmoved into or out of the state, etc. 76

• Personal Income measures how much personal income was generated per person, on theaverage. It equals the total amount of income generated in that state divided by themid-year population of that state. For South Dakota, this was $25,958 in 2000.

• The Child Poverty Rate measures what percent of children in each state are classifiedas living in poverty. The rate for South Dakota was 14% in 2000.77

(a) The four scatter-plots above each show a relationship between one of the given variablesand PFIH. For each of these relationships, consider the plots, and say whether thereseems to be a positive association, a negative association, or no association, and indicatewhy you decided as you did.

(b) The table below shows the correlation coefficients for each of the four relationships,along with the squared coefficients r2.

Variable r2 rPer Capita State Tax Revenue 0.076428 -0.27646Percent Drop in Welfare Rolls 0.012054 0.109792Per Capita Income 0.277138 -0.52644Child Poverty Rate 0.392424 0.626437

i. According to these coefficients, which relationships show a positive association?Which relationships show a negative association?

ii. Which of the 4 relationships shows the weakest association (one that seems to beclosest to having no association)?

iii. Which relationship shows the strongest positive association? Do you think it issurprising that this variable has the strongest positive association, or would youhave expected this? Explain briefly.

iv. Which relationship shows the strongest negative association? Do you think it issurprising that this variable has the strongest negative association, or would youhave expected this? Explain briefly.

(c) When the welfare reform legislation was passed in the mid-1990’s, many critics predictedthat there would be adverse consequences, including an increase in hunger and foodinsecurity. The argument was that as individuals and families were dropped from welfareprograms, many would have trouble making ends meet, including procuring enough food.According to the information above, does this prediction seem to have proven true ornot? Explain as completely as you can why you think so.

75Data is from the U.S. government Census Bureau, http://www.census.gov/govs/statetax/00staxss.xls76Welfare data is from http://www.infoplease.com/ipa/A0774474.html77Income and poverty data found at http://www.hungerfreeamerica.com/facts/statistics/index.cfm

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5. Ozone Ouch.78 The following table gives the maximum area of the ozone hole over Antarc-tica for most years since 1980.79 The area is in millions of square kilometers.

Year Area Year Area(Millions sq. km) (Millions of sq. km)

1980 3.27 1993 24.0171981 3.15 1994 23.4291982 10.8 1995 NA1983 12.24 1996 26.961984 14.65 1997 25.131985 18.79 1998 28.211986 14.37 1999 26.091987 22.45 2000 30.311988 13.76 2001 26.521989 21.73 2002 21.741990 21.05 2003 28.511991 22.60 2004 22.761992 24.90 2005 26.77

For purposes of comparison, the United States is a little over 9 million square kilometers,and North America is a little over 25 million square kilometers.

(a) Let time be measured in years since 1980. So 1980 is represented by t = 0, 1990 byt = 10, etc. Find the equation for the regression line where years since 1980 is the inputand the area of the ozone hole is the output.

(b) Using your equation from part (a), compute the area of the ozone hole for 1980, 1987,1994, and 2001. For each year, find the difference between your result and the actualarea of the hole given in the table. What does this tell you about your regression line?

(c) What is the correlation for your regression line? What information does this give youabout how well the line fits the data?

(d) Using your equation from part (a), predict the area of the hole in the year 2015. Howconfident are you about the accuracy of your answer? Why?

(e) Notice that the trend in the graph seems to be steeper before 1990 than after 1990.Suppose that you were doing this research in 1990 and only had the data through thatyear. Find the regression line and correlation coefficient for this smaller data set. (Oneway to do this in Fathom is to create a copy of the case table, and delete the years youdo not want to include). Let’s call this second linear model f2(x)

i. Graph y = f2(x) together with the full data set. This can be done in Fathom byadding a movable line, and adjusting it until the equation approximates the one youhave for y = f2(x). How well would f2 have predicted the area of the ozone hole forthe years 1990 through 1995? 1996 through 2001?

ii. Does this new information change your answer to part (d)?

78adapted from Understanding Our Quantitative World by Janet Andersen and Todd Swanson79Data from http://www.theozonehole.com/ozoneholehistory.htm


6. Inequality and Poverty. The table below gives the Poverty Rate80 as well as the GiniNumber (or Coefficient) for 57 countries. The Poverty Rate is the percentage of people inthat country considered to be living in poverty. As noted in chapter three, the Gini Numberis a measure of income inequality, and can be considered in some sense the “percentage ofinequality” in a country.81

The inequality data were collected by Deininger and Squire in the early 1990’s. The countrieslisted are those for which Deininger and Squire had both poverty data and Gini Coefficients.If you are curious, the Gini Number for the USA is estimated at 37.2. One question onemight consider is “Do poorer countries have higher levels of inequality?”

The data is available in the Fathom File Inequality and Poverty Data

Country Pov Rt. Gini No Cont. Country Pov Rt. Gini No Cont.Bangladesh 49.8 28.27 Asia Malawi 65.3 62 AfricaBelarus 41.9 28.526 Europe Mali 63.8 54 AfricaBulgaria 12.8 27.7967 Europe Mauritania 46.3 37.8 AfricaBurkina Faso 45.3 39 Africa Mauritius 10.6 36.69 AfricaChile 17 52.35 SA Mexico 10.1 50.31 NAChina 4.6 29.9662 Asia Moldova 23.3 34.43 EuropeColombia 64 50.0233 SA Morocco 19 36.2167 AfricaDjibouti 45.1 38.1 Africa Nicaragua 47.9 50.32 NADom. Rep. 28.6 49 NA Niger 63 36.1 AfricaEcuador 35 43 SA Nigeria 34.1 39.31 AfricaEgypt 16.7 32 Africa Pakistan 32.6 31.15 AsiaEl Salvador 48.3 44.77 NA Paraguay 21.8 39.8 SAEstonia 8.9 37.224 Europe Peru 49 38.42 SAEthiopia 44.2 44.2 Africa Philippines 36.8 45 AsiaGhana 39.5 33.94 Africa Poland 23.8 26.6845 EuropeGuinea 40 40.4 Africa Romania 21.5 25.33 EuropeGuin. Bissau 48.7 56.12 Africa Russian Fed. 30.9 30.53 AsiaGuyana 35 40.22 SA Senegal 33.4 54.12 AfricaHonduras 53 53.726 NA Sri Lanka 25 30.1 AsiaHungary 17.3 27.8743 Europe Tanzania 35.7 51.37 AfricaIndia 28.6 31.992 Asia Thailand 13.1 50.264 AsiaIndonesia 27.1 33.1371 Asia Tunisia 7.6 40.62 AfricaJamaica 18.7 39.825 NA Uganda 55 39.6933 AfricaJordan 11.7 40.66 Asia Ukraine 31.7 25.71 EuropeKazakhstan 34.6 32.67 Asia Venezuela 31.3 49.12 SAKenya 42 54.39 Africa Vietnam 50.9 35.71 AsiaKyrgyz Rep. 64.1 35.32 Asia Zambia 72.9 49.1033 AfricaLaos 38.6 30.4 Asia Zimbabwe 34.9 56.83 AfricaMadagascar 71.3 43.44 Africa

(a) Create a scatterplot of Gini No versus Pov Rate. Visually, how would you describe theassociation between these variables? Is it positive or negative, weak or strong?

(b) Find the correlation coefficient for this relationship. What does this additional informa-tion tell you about the association?

80The data were downloaded from the World Bank website at http://www.worldbank.org/research/growth/dddeisqu.htm.81Recall, for example, that if everyone in the country made exactly the same amount of money, the Gini number

would be 0. If one person in the country had all the income, and everyone else had none at all, the Gini numberwould be 100 (for 100% inequality).

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(c) Find the linear regression line for this data. Graph this line on the scatter-plot. Explainwhat the slope and y-intercept of this line tell you in terms of the two variables.

(d) Use the linear regression equation to predict the Gini Coefficient for countries with 0%poverty, 50% poverty, and 100% poverty. Do these predictions seem reasonable? (Hint:Do your predictions correspond to the approximate values you’d get from the graph?)

(e) It is possible that certain regions or continents might show a weaker or stronger rela-tionship between these variables than the data as a whole.

i. Construct a scatter-plot just for African countries. You can do this by creating aFilter in Fathom, or by creating a duplicate table and deleting the unwanted cases.Does the African data seem to show a stronger or weaker association than the dataas a whole?

ii. Find the linear regression line and correlation coefficient for the African data. Howdo these compare with those from the data as a whole?

iii. Find the linear regression line and correlation coefficient for the Asian countries.How do these compare with those from Africa? Do you find anything surprising?

(f) Look over the list of countries shown. Are there any particular types of countries orregions that seem to be left out?

(g) Finally, let’s investigate which continent, Africa or Asia, has more inequality, as mea-sured by the Gini Coefficient. Create one histogram for Gini Coefficients, includingseparate bars for Africa and Asia. An easy was to do this in Fathom is to create a graphwith Continent as the Y attribute and Gini No as the X attribute. This will give you all5 continents, and you can select Africa and Asia using a Filter. What do you conclude?

7. Comparing Car Models. The data in the table below represent 38 car types of 7 differentmakes and 19 different models and variations for the 2005 model year. The data is fromwww.autos.msn.com. All models have automatic transmissions. The average price is basedon the Kelley Blue Book price for the 2004 model year. City Mileage, Highway Mileage, andGreenhouse Gas Emissions ratings are as determined by the EPA. Gas emissions is given intons of carbon dioxide per year under normal driving conditions. Engine Size is in liters.Passenger Room and Luggage Room are given in cubic feet.


City H’way Carbon Engine Passenger Lugg.Make Model Price mpg mpg Emiss. Size Room RoomHonda Civic 15325 29 38 5.9 1.7 91 10Honda Accord (2dr) 22622.5 24 34 7 2.4 103 14Honda Accord (4dr) 22622.5 24 34 7 2.4 103 14Toyota Avalon 23500 22 31 7.6 2.2 107 14Honda Accord (2dr) 22622.5 21 30 7.8 3 103 14Honda Accord (4dr) 22622.5 21 30 7.8 3 103 14Toyota Corolla 15667.5 30 38 5.7 1.8 84 14Honda Civic 15325 29 38 5.9 1.7 91 10Ford Crown Victoria 27475 18 25 9.2 4.6 111 21Toyota Matrix 4x2 16855 28 34 6.2 1.8 94 22Toyota Matrix 4x4 16855 28 34 6.2 1.8 94 22Toyota Celica 18600 29 36 6 1.8 78 17Toyota Camry 21875 24 34 6.9 2.4 102 17Chevrolet Cobalt (4dr) 17527.5 24 32 6.7 2.2 87 14Ford Taurus SW 13875 19 25 8.8 3 109 39Toyota Camry 21875 21 29 8 3.3 102 17Ford Taurus 22165 20 27 8.4 3 105 17Toyota Camry 21875 20 28 8.2 3 102 17Chevrolet Monte Carlo 25317.5 21 32 7.6 3.4 96 16Toyota Corolla 15667.5 30 38 5.7 1.8 84 14Chevrolet Aveo 11160 26 34 6.5 1.6 91 12Saturn Ion (2dr) 11900 24 32 6.7 2.2 91 15Saturn Ion (4dr) 11900 24 32 6.7 2.2 91 15Ford Focus 15842.5 26 32 6.8 2 94 15Chevrolet Malibu Classic 21630 25 34 6.7 2.2 99 16Toyota Camry 21875 24 34 6.9 2.4 102 17Honda Accord (4r) 22622.5 24 34 7 2.4 103 14Chevrolet Cobalt (4dr) 17527.5 24 32 6.7 2.2 87 14Dodge Stratus (4dr) 21845 22 30 7.6 2.4 94 16Toyota Camry 21875 21 29 8 3.3 102 17Honda Accord (4r) 22622.5 21 30 7.8 3 103 14Dodge Stratus (4dr) 21845 21 28 8.1 2.7 94 16Toyota Camry 21875 20 28 8.2 3 102 17Dodge Stratus (2dr) 21845 21 28 8 2.4 86 16Dodge Stratus (2dr) 21845 20 28 8.3 3 86 16Dodge Neon 13750 25 32 6.8 2 90 13Chevrolet Cavalier (2dr) 14017.5 24 34 6.7 2.2 92 14Chevrolet Cavalier (4dr) 14017.5 24 34 6.7 2.2 92 14

(a) How closely related is city mileage and highway mileage for these car models? To answerthis question, create a scatterplot and find the linear regression line and correlationcoefficient for these two variables.

i. What is the slope of your linear regression equation and what does this slope meanin terms of city and highway mileage?

ii. What is the y-intercept of your linear regression equation and what does this slopemean in terms of city and highway mileage?

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(b) Let’s consider greenhouse gas emissions.i. What is the average amount of carbon dioxide emissions for these models?ii. Of the variables city mileage, highway mileage, average price, engine size, and pas-

senger room, which is most highly correlated with carbon dioxide emissions? Ex-plain how you know.

iii. Based on these data, in order to reduce greenhouse gas emissions by one ton peryear, how much would you need to increase your city gas mileage? Explain how youknow.

(c) Do these data support the statement “consumers who wish to reduce their contributionto greenhouse gas emissions are going to have to pay more for their vehicles”? Considerhow you can use the data to answer this question and justify your answer.

(d) This past fall, the “big three” U.S. automakers, Ford, Chrysler (Dodge), and GeneralMotors (makers of Chevrolet and Saturn) lobbied congress and the President for “bailoutloan funds.” These car makers were criticized for not producing the fuel efficient, “green”vehicles that the times demand. Based on these data, is it fair to say that U.S. car modelsare less fuel efficient and less environmentally friendly than cars produced by foreign carcompanies like Honda and Toyota? Be sure to justify your answer.

8. Miles Per Gallon.82 With the significant increases in the cost of gasoline that have occurredsince 2005, owning a vehicle is becoming even more expensive. The purchase price is thebiggest one time cost. Continuing costs include not only gasoline, but maintenance, andinsurance. The cost of gasoline is related to the miles per gallons averaged by the car. Moreefficient cars will reduce this cost, sometimes dramatically (assuming you continue to drivethe same number of miles).

(a) Suppose you were considering buying a car in the fall of 2008 or 2009, and were choosingbetween a Honda Accord and a Ford Taurus. A 2-door Honda with a 2.4 liter engine andan automatic transmission costs about $23,725 MSRP (manufacturers suggested retailprice), gets 21 miles per gallon in the city and 30 on the highway. A 4-door Taurus witha 3.5 liter engine has a MSRP of $25,670, gets 18 mpg in the city and 28 on the highway.Assume most of the miles you drive are highway miles, and the cost of gasoline is $1.69per gallon. We’ll also assume maintenance and insurance costs are roughly the same forboth models.83

i. Write linear equations for the Accord and the Taurus where the input is miles andthe output is the cost for owning and operating the vehicle. Include the initial costof buying the vehicle and gasoline costs but ignore maintenance and insurance costs.

ii. Graph the equations on the same axis system, using a window that covers how longyou expect the cars to last, and shows the intersection point of the two graphs.Indicate the window used and the intersection point.

iii. What is the physical meaning of the intersection point?(b) Some energy experts consider $5 per gallon gasoline a realistic possibility in the near

future. Redo part (a) under the alternate assumption that gas is $5 per gallon.(c) Some metropolitan areas are looking at light rail transit and other forms of mass trans-

portation as a way to reduce traffic, pollution, and gasoline usage.i. How much would it cost to run a car that is driven 15,000 each year and gets 24

mpg on the average. Assume gas is $5 per gallon.ii. How much would it cost to run 200,000 such cars for a year?iii. How much money would be saved (say, to help pay for mass transit) if these 200,000

cars cut their mileage by two-thirds, to 5,000 miles from 15,000?

82adapted from Understanding Our Quantitative World by Janet Andersen and Todd Swanson83Information on these two models is from www.kbb.com, the Kelley Blue Book site

6. Exponential Functions

The single biggest problem of the human race is its inability to understand the exponen-tial growth function—Albert Bartlett

In the last chapter, we considered linear functions, our first “family.” We saw that linearfunctions provide good models for relationships where we know there is a constant rate of change,or where the scatterplot of the data seems to follow a roughly linear pattern. Of course, lots ofrelationships do not have a constant rate of change, and lots of of scatterplots or line graphs are notvery linear. Recall, for example, Figures 1 and 2 of Chapter Two which depicted Hubbert Curvesof energy productions data, or the Lorenz curves we used in introducing the Gini Coefficient inChapter Two.

In this chapter and the next four, we will discuss several more families of functions, considersome of the basic characteristics that define these families, and see how they can be useful inmodeling nonlinear patterns of data. There are certainly many families of nonlinear functions, but,as with linear functions, we will consider the simplest first and progress to some more complicatedexamples. In this chapter, we will consider the exponential family of functions.

Exponential Functions Symbolically

Symbolically, linear functions are those which can be written in the form y = mx+b. We calledm the slope, and b is the y-intercept. To define a particular linear function, one need only specifyparticular values for m and b. We call m and b parameters. Although designated by letters,parameters are not considered variables (x and y are the variables here), but rather are consideredunspecified constants.

In linear functions, the independent variable x does not involve an exponent, unless you wantto write it as x = x1. Exponential functions, and also power functions which will be consideredin detail in the next chapter, will involve exponents that are not equal to 1. Recall that, in anexpression of the form br, b is called the base and r is called the exponent. The appendicescontain some additional background and examples related to exponents and roots that you canrefer to as needed throughout this section.

• Exponential functions are those which can be written in the form y = a · bx. The variableis an exponent and the base is a constant.

• Power functions, which we define here but which will be discussed in detail in ChapterEight, are those which can be written in the form y = axb. The variable is a base and theexponent is a constant.

In both cases, a and b are parameters,84 just as m and b are parameters for the family of linearfunctions y = mx + b. The families are defined by which operations are performed on parameters,and which on the independent variable. In exponential functions, the independent variable is theexponent, and we denote the base by the parameter b.

Here are some examples of exponential functions:

y = 3 · 2x f(x) = 2.15(0.87)x p(t) = 6.2(1.016)t

Here are some examples of power functions:

y = 3 · x2 f(x) = 2.15(x0.87) p(l) = 10.5(l)1.5

84Which letters are used to denote the parameters is a somewhat arbitrary choice. We could have written y = bax

so that a was the base. The important thing is to substitute consistently into the general equation given for a family.

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128

Example 1. Here are a few more functions in symbolic form. Identify which of the threefamilies, linear, exponential, or power, they belong to, or if they belong to none of the three (thereis one of these!).

1. y = 0.0345x− 5

2. y = 10500(x2.3)

3. y = 2x(2.3 + x)

4. y = 2x

5. y = 3.4(1.032x)

Solution: Item 1 is linear, item 2 is a power function, and items 4 and 5 are exponentialfunctions. Item 3 is none of these.85

Example 2. Sometimes, functions might appear “in disguise,” so that you might have to do alittle algebraic manipulation to recognize them. Which do you think are linear? Which exponentialor power?

1. y = 2x−0.15

2. y = 0.02(x− 0.15) + 2

3. y = (3x)2

4. y = 7(100.023x)

5. y = 4 5√

x3

Solution: Let’s go through each of these five examples in turn.

1. Let’s try simplifying this, using properties of exponents.

y = 2x−0.15 = 2x+−0.15 = 2x2−0.15 ≈ 2x(0.901) = 0.901 · 2x

This is an exponential function with parameters a ≈ 0.901 and b = 2

2. This is a linear function.

y = .02(x− 0.15) + 2 = 0.02x− 0.02(0.15) + 2 = 0.02x− 0.003 + 2 = 0.02x + 1.997

The slope is 0.02 and the y-intercept is 1.997.

3. This is a power function. y = (3x)2 = 32x2 = 9x2, and so is of the form y = axb with a = 9and b = 2.

4. You might guess this is also exponential.

y = 7(100.023x) = 7((100.023)x) ≈ 7(1.0544x)

In fact, it is an exponential functions with base 10.023 or about 1.0544.

85In fact, this function is a quadratic function, which is another family we will eventually consider

6. Exponential Functions 129

5. We have not considered root functions as yet. However, you may know that roots can bechanged to exponents.86 For example,

3√

x7 = x7/3

In our case, we have

y = 4 5√

x3 = 4x3/5

which is a power function.

In considering these five functions from Example 2, we used basic properties of exponents, thedistributive law, and in the last, the definition of fractional exponents. We also used a calculatorto get decimal approximations in two cases. Be sure you understand the steps involved in theseexamples, in case we run into any more functions “in disguise!”

Exponential Functions in Graphical Form

In the previous chapter, we saw that any linear function had a symbolic representation ofthe form y = mx + b and this corresponded to the graph of the function being a straight line.We saw that if m was positive, the graph was increasing, and if m was negative, the graph wasdecreasing. We also know that b is the value of the y-intercept. How can we characterize thegraphs of exponential functions, and how are these characteristics related to the parameters in thesymbolic representation?

In these examples, and the ones that follow, we will be interested not only in the generalcharacteristics that are true for all functions in the family of exponential functions, but also howthese characteristics are affected by the parameters. We will consider these questions for the otherfamilies of functions we consider, as well as the similarities and differences between functions indifferent families.

Figure 1 shows the graphs of exponentials functions f(x) = 3 · 1.5x, g(x) = 3 · 2x, and h(x) =3 · 2.5x.

h(x)

g(x)

f(x)

Figure 1: Graphs of f(x) = 3 · 1.5x, g(x) = 3 · 2x, and h(x) = 3 · 2.5x

86As noted above, see appendices for more information on exponents and roots

130

The graphs are fairly simple, so in some sense there may not seem to be much to notice. Wecan say that all the graphs are increasing, and they are all concave up. We might also notice thatthe graphs all have the same y-intercept of 3, and that this is the the same as the parameter a inthe symbolic form y = abx! Why is this? Recall that the y-intercept is the y-value of the functionswhen x = 0. In the case of an exponential function y = abx, if x = 0, we have

y = ab0 = a · 1 = 1

since any non-zero number to the zero power is equal to 1.Do the graphs have any x-intercepts? At least in the window we see, there are none. In fact,

there are no x-intercepts because if b is any positive number, bx will also be positive for any valueof x.

How does the base b affect the graphs? If we look on the right side of the graph (where x > 0),we see that the larger the base, the steeper the graph. For the function h, b = 2.5 which is thelargest base and h has the steepest graph. On the other hand, to the left of the origin (x < 0),all the graphs are very flat. As you go further to the left (in other words, as x becomes more andmore negative) the graphs seem to become flatter and the y-values become closer to 0. We saythat y approaches 0, as x goes to negative infinity, and that y = 0 is a horizontal asymptotefor the graph. An asymptote is a straight line which a graph gets closer and closer to, oftentimeswithout ever touching. Exponential functions get closer and closer to the x-axis without touching.

Why does this happen? Recall that, if n is positive, b−n means the same as 1/bn. In otherwords, negative exponents mean “take the reciprocal.” When we let x be a negative number, thenbx is really the reciprocal of b to the corresponding positive power. For example, if x = −3 andb = 2, bx = 2−3 = 1/23 = 1/8, which is fairly close to zero. If the base b is bigger than 1, thenthe “more negative” the exponent is, the larger the denominator in the resulting fraction, and thesmaller the resulting fraction.

Example 3. In Figure 1, we looked at three functions of the form y = abx, but in all cases, bwas bigger than 1. What happens if we use b where 0 < b < 1? On your graphing calculator (or acomputer software program), graph several examples of exponential functions where the parameterb is between 0 and 1. Write a short paragraph or two, similar to what we did above, describing thecharacteristics of these exponential functions and how the parameters a and b affect the graphs.You might also experiment with different graphing windows. The main idea is for you to “playaround” with examples of these types of exponential functions until you feel like you understandhow they work.

Solution: Figure 2 shows some possible examples for graphs of y = abx with 0 < b < 1. Youmay or may not have graphed any of these particular functions.

These graphs are also concave up, but instead of increasing, they are all decreasing. They aresteeper on the left and flatter on the right, and these functions seem to level off at y = 0 as xbecomes larger and larger (in other words, as x goes to infinity). As in the previous case, y = 0seems to be a horizontal asymptote. Also as before, the parameter a represents the y-intercept.

Finally, note that since exponential functions are either increasing throughout their domain, ordecreasing throughout their domain, every exponential function is a one-to-one function. Recallfrom Chapter Four that this means that if f(a) = f(b), then a must equal b. So, for example, if2a = 2b, this can only happen if a = b.

Numerical Representation of Exponential Functions

If we have a function given in numerical form, we can determine if it is linear by seeing if ithas a constant rate of change. In other words, for each one unit increase in x, we can check to seeif we always have the same increase in y. Alternatively, we can see if any two points selected fromthe table give us the same slope.

Is there a way we can tell if a function given in numerical form is exponential?


Figure 2: Graphs of f(x) = 0.8x, g(x) = 2(0.6)x, and h(x) = 4(0.5)x

YES!!

While a linear function g(x) has a constant rate of change, equal to g(x + 1)− g(x), we will seethat an exponential function f(x), has a constant growth factor, which is equal to f(x+1)

f(x) . Forexample, let E denote the exponential function E(x) = 2x (parameters a = 1 and b = 2). Somevalues for E(x) are given in Table 1. Notice that if we take any y-value divided by the previousy-value in the table, the result is 2. In other words E(x+1)

E(x) = 2 for any x. The growth factor of thisfunction is 2. Alternatively, we could write E(x + 1) = 2 · E(x), so that taking any y-value timesthe growth factor of two gives the next y-value in the table. Thus, the result of increasing x by 1is that y gets multiplied by the growth factor 2.

In other words, addition is to linear functions as multiplication is to exponential func-tions. Table 1 shows an example of an exponential function with growth factor 2 andan linear function with slope 2 for comparison.

Table 1: Exponential function E(x) = 2x, and linear function L(x) = 2x + 1.

x 0 1 2 3 4 5 6 7 8 9 10E(x) 1 2 4 8 16 32 64 128 256 512 1024L(x) 1 3 5 7 9 11 13 15 17 19 21

Notice how the exponential function begins to increase very, very rapidly as compared to the linearfunction’s constant growth rate.

Let’s generalize this. If E(x) = a · bx, then we can create a table for the function as always,except that the values will involve the parameters a and b.

Notice that if we multiply any y-value by the growth factor b, we get the next y-value (ab2 · b =ab3, ab3 · b = ab4, etc.). So, the base b is the growth factor.

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Table 2: Exponential function where E(x) = abx.

x 0 1 2 3 4 5 6 · · · n

E(x) a ab ab2 ab3 ab4 ab5 ab6 · · · abn

Example 4.For each of the following tables, say which are tables for exponential functions, which for linear

functions, and which for neither.

1.x −2 −1 0 1 2

f(x) 1 4 16 64 81

2.x −2 −1 0 1 2

f(x) 12

14

18

116

132

3.x −2 −1 0 1 2

f(x) 0.8 0.4 0 −0.4 −0.8

4.x −2 −1 0 1 2

f(x) 0.3 0.45 0.675 1.0125 1.51875

Solution:Notice that, in each case, the x-values in the table increase by 1 each time, so we can check

whether the functions are exponential by calculating the ratios f(x+1)f(x) .

1. This function looks like it may be exponential since 41 = 16

4 = 6416 = 4, but since the ratio

8164 = 1.265625 6= 4 the function is not exponential. If we changed the value 81 to 4 ·64 = 256,then we would have an exponential function.

2. This function is exponential with common ratio 1/2.

3. This function is not exponential since 0.40.8 = 1

2 6= 00.4 . Instead, this is actually a linear function

where the slope is m = −0.4. In addition, note that 0 is an output for this function, whichcan never be true for an exponential function.

4. Calculating the ratios, we get a common ratio of 0.450.3 = 1.5 (you can check the others via

calculator). So, this is an exponential function.

The Number e

Any positive number b 6= 1 can be used as the base for an exponential function f(x) = abx.Some bases are used more often than others, however. The most frequently used are probablyb = 10 and b = e. The latter is particularly used by many software packages.

The number e is approximately 2.71828. Like the symbol “π” which stands for the numberused in formulas for the area and circumference of circles (π ≈ 3.14159), the symbol “e” is usedsimply as a short hand for a rather complicated number that we do not want to have to writeout constantly. Like π, the number e is an irrational number. You may recall that this means it


is represented by an infinite non-repeating decimal. In practice, calculations with e are done viacalculators; nearly any calculator these days has an ex button.

Additional background on e as well as some history on why it has come to be important areprovided in the appendices. The main importance for us will be the use of e in software packages,and the interchangeability between base e and other bases. Also, a useful fact is that

(1 + r)t ≈ ert

for “small” values of r. Recall that in the expression (1 + r)t, (1 + r) is the growth factor and sor represents the percentage growth rate. Similarly, in the expression ert, the r is also referred toas the percentage growth rate. Table 3 gives some values of these two expressions for various r forpurposes of comparison.

Table 3: Comparison of (1 + r)t and ert.

r 0.01 0.03 0.05 0.1 0.15 0.2(1 + r)t 1.0510 1.1593 1.2763 1.6105 2.0114 2.4883

ert 1.0513 1.1618 1.2840 1.6487 2.1170 2.7183

If you have a function of the form f(x) = a · ert, you can convert it to the form a · bt simply bycalculating er. For example, 2e0.035t = 2(e0.035)t ≈ 2(1.03562)t.

Non-linear Regression Models

Many spreadsheets, statistical software packages, and graphing calculators, including TI graph-ing calculators, have built in procedures for finding exponential and other non-linear regressionmodels. Typically, exponential and logarithmic regression are the most popular, but you may alsofind power regression, quadratic regression, and other regressions available, depending on whattechnology you are using. These non-linear regression features give you the function from theparticular family of functions (eg. exponential) that “best” fits your data. We will discuss thedetails of how the technology is finding these models later, as well as how the definition of “best”may be somewhat different than the definition for linear regression, but for now, let’s just do a fewexamples using these capabilities. More detailed explanations on how to find non-linear regressionsusing a TI graphing calculator, as well as how to find exponential and logarithmic regressions withEXCEL, are give in the Appendices.

For example, in Chapter Four87, we considered two different models for damages due to floodsin the U.S. One was a linear model, and the second was an exponential model (although we didnot use the term “exponential” at that point). The linear model was found using linear regression,as in Chapter Five. The exponential model, D1(t) = 0.741(1.0168t), was found using exponentialregression.

Notice that the model graphed with the data is slightly concave up and is increasing, consistentwith an exponential function. The exponential model is given by Excel in the graph, and isy = .7414(e.0167t). Calculating e.0167, we get 1.0168, so this equation is equivalent to the modelD1(t) = 0.741(1.0168t) originally given in Chapter Four.

Most mathematical technologies will also give you a correlation coefficient for non-linear re-gressions. As with linear regression, this is a measure of how well the function fits the data.Unfortunately, the correlation coefficients for the exponential regression and other non-linear re-gressions are not based on the sum of squared errors between the given equations and the data,as they are for linear regression. This means, for example, that if linear regression gives you acorrelation coefficient of r = .9 and exponential regression gives a value of r = .95, you cannotconclude on this basis alone that the exponential regression function fits the data better than the

87See Example 3 and preceding.

134

US Flood Damages Adjusted for Inflationy = 0.7414e0.0167x

$0.00

$4.00

$8.00

$12.00

$16.00

$20.00

0 20 40 60 80 100

Year from 1900

Dam

ages

in B

illio

ns

of

$

Figure 3: U.S. Flood Damages Data in Real Billions of Dollars with Exponential Regression Model

linear. However, if you have two different exponential models, you can compare them to each otherusing the correlation coefficients.

Application: Compound Interest

Nearly all fixed rate investments today use compound interest. Compound interest simplymeans that you are payed interest at fixed intervals, typically every month, and once the interestis paid, it becomes part of the principal that is invested and so earns interest along with the restof the principal.

For example, suppose way back when you were born, your grandmother placed $2000 in anaccount which paid a 6% annual interest rate compounded monthly. Since the 6% is an annualrate, the monthly rate would by 6%/12 = 0.5%, and each month her account would be creditedwith interest equal to 0.005 times the previous month’s balance. The following table shows how theaccount would grow over the first few months. We have shown some of the expressions unsimplifiedto illustrate the pattern of how compound interest works.

Beginning Interest EndingMonth Balance Earned Balance

1 2000 0.005 · 2000 = 10 2000 + 0.005 · 2000 = 2000(1 + 0.005) = 20102 2010 0.005 · 2010 = 10.05 2010 + 0.005 · 2010 = 2010(1 + 0.005) = 2020.053 2020.05 0.005 · 2020.05 = 10.10 2020.05 + 0.005 · 2020.05 = 2020.05(1 + 0.005) = 2030.154 2030.15 0.005 · 2030.15 = 10.15 2030.15 + 0.005 · 2030.15 = 2030.15(1 + 0.005) = 2040.30

Table 4:

Notice that we can write the ending balance each month as 1.005 times the beginning balance.If we consider months as the input, increasing the input by 1 results in the output being multipliedby 1.005. This is an exponential function with base 1.005. In fact, using t as the input variable,the function B(t) which gives the balance at the end of t months is given symbolically by


B(t) = 2000(1.005)t

which fits the form of our exponential function family with a = 2000 and b = 1.005.Knowing the symbolic form for this function, we can find out how much money she will have to

give you (if that’s what she decides to do) on your 18th birthday. There are 12 · 18 = 216 monthsin 18 years. So, the account would be worth B(216) = 2000(1.005216) ≈ $5873.53.

Now, what if we wanted to consider another compound interest situation, with an in-terest rate different than 6% compounded monthly?

Well, we could use the same logic as in our original example for any interest rate. We could evenallow for other compounding periods, besides monthly compounding, as well as allowing for achange in the amount invested. In our original example, 0.005 was the monthly interest rate, $2000was the amount invested, and t denoted the number of months. To generalize to any rate, anyamount invested, and any compounding period, we will let i denote the periodic interest rate,P the amount invested, and t the number of periods. Using this notation, our general compoundinterest formula is

B(t) = P (1 + i)t

For example, with an account with an annual interest rate of 8% compounded quarterly (four timesper year), we would have i = 0.08/4 = 0.02 and our formula would be

B(t) = P (1.02)t

where t would be the number of quarters we leave the money in the account. If we wanted to leavethe money in for 18 years, this would be 4 · 18 = 72 quarters, so we would let t = 72.

Example 5. Suppose you invest $5000 at 6% compounded monthly. How much will you havein 8 years?

Solution: Since you are compounding monthly, your periodic interest rate is i = 0.06/12 =0.005. In eight years, there are 8 ·12 = 96 months. So your balance will be B = 5000(1+0.005)96 ≈$8070.71.

Application: Inflation and Real Dollars

You have probably heard your grandparents saying something like, “In my day, I could go tothe movies for 50 cents!” However, what they usually fail to tell you is that 50 cents was wortha lot more in their day than it is now. The concept of money changing in value through time isknown as inflation. Table 5 shows the inflation factor for every 5 years from 1935 to 2000, andthen 2001, 2002, and 2003. The inflation factor represents how much it would cost in that year tobuy an “average” set of items that cost $1 in 2003.88

Year 1935 1945 1950 1955 1960 1965 1970 1975Inflation Factor 0.074 0.098 0.131 0.146 0.161 0.171 0.211 0.293Year 1980 1985 1990 1995 2000 2001 2002 2003Inflation Factor 0.448 0.585 0.711 0.829 0.936 0.963 0.978 1.000

Table 5: Inflation factors from 1935 to 2003.

So, for example, say it cost $1 in 2003 to buy an apple, a can of soup, and a pencil. Then,because of inflation, we can see from the table that the same set of items should have cost only 13

88Inflation factors from www.infoplease.com.

136

cents in 1950, 16 cents in 1960, etc. For a set of items that cost several dollars in 2003, to find thecost in any other year N you can multiply the 2003 cost by the inflation factor in year N to get theestimated total cost for the items in that year. So, if you bought a bag of apples, a pair of jeans,and a box of laundry soap for a total of $35 in 2003, the estimated cost for those same items in1935 would be 0.074 · 35 = $2.59. The idea is that, on average, things cost more today than theydid years ago. The inflation factor keeps track of this average rise in the price of consumer goods.

Where did these inflation factors come from? Recall in Chapter Three that we introduced theidea of an index, and even mentioned the Consumer Price Index as one example. Each month,the government surveys retail outlets all over the country and finds out how much more a “typicalbasket” of consumer goods costs this year than it did last year. The goods that are selected to be apart of this “basket” are the components of the index, and you can imagine that there might havebeen some changes in these components over time.89 The percentage increase in the ConsumerPrice Index (CPI for short) is the standard measure for inflation. When you hear on the news thatthe inflation rate was 3.2% last year, this means that the CPI is 3.2% higher this year than it waslast year.

The inflation factors are calculated so that the percent increase in the inflation factor from oneyear to the next is the same as the inflation rate. So for example, if we compare the inflationfactors for 2002 and 2001 in Table 5, we have 0.978/0.963 = 1.015576. This represents a 1.5576%increase, and this increase is the same as the percentage increase in the CPI from 2001 to 2002,which is the inflation rate.90

The difference between the CPI and inflation factors is that we pick a particular year to set theinflation factor to 1, representing $1. This is called the reference year (sometimes also called thebase year). We could have picked any year we wanted to be the reference year. All you need todo to compute the inflation factors from the CPI is pick your reference year, and then divide allthe CPI values for the years you want by the CPI for the reference year.

In Table 5 above, we picked 2003 as the reference year. In Table 6 we have changed the referenceyear to 1980. Table 6 shows the CPI values for the given years, and the inflation factors are allcalculated by dividing the CPI in the given year by the CPI in 1980, namely 82.4.

Year 1935 1945 1950 1955 1960 1965 1970 1975Inflation Factor 0.165 0.219 0.292 0.326 0.359 0.382 0.471 0.654CPI 13.7 18.0 24.1 26.8 29.6 31.5 38.8 53.8Year 1980 1985 1990 1995 2000 2001 2002 2003Inflation Factor 1.000 1.306 1.587 1.850 2.089 2.149 2.183 2.232CPI 82.4 107.6 130.7 152.4 172.2 177.1 179.9 184.0

Table 6: CPI and Inflation factors from 1935 to 2003, with reference year 1980.

Currently, the CPI is calculated so that the average for the years 1982 through 1984 is 100.This is a somewhat arbitrary choice, just like our decision on which year to use as the referenceyear for inflation factors. The important thing is that the percent increase in the CPI reflect theaverage increase in prices of all items as accurately as possible. If we wanted, we could use inflationfactors with any reference year we wish to keep track of inflation instead of the CPI, since theywill all have the same percentage increases from any given year to the next year.

Example 6. Find an exponential model for the inflation factors in Table 6. Does your modelseem to fit the data well? What does the percentage growth rate represented by this model tellyou?

Solution: Using exponential regression on the data in Table 6 and letting t = 0 represent 1935,we get an exponential model of y = .142(1.0421t). The correlation coefficient is r = 0.99. A

89in 1900, you might have included supplies for your horse, and today you might include items like DVD’s whichdid not exist even a few years ago.

90The rate would probably be reported as 1.6% or 1.56%.


scatterplot of the data along with the exponential model is shown in Figure 4.Inflation Factors0.000.501.001.502.002.50

1935 1945 1955 1965 1975 1985 1995 2005Figure 4: Graph of inflation factors, reference 1980, with exponential model

The model seems to fit the data reasonably well, although some of the errors might be consideredsignificant. The largest error seems to occur in 1985 and to be around 0.15. In fact, the modelwould predict an inflation factor of 0.142(1.042150) ≈ 1.16, versus the actual value of 1.306. Thismeans the model would underestimate what you would have needed to spend in 1985 by 15 centsfor every $1.30 spent.

The trend for the last 3 years seems to show a slower increase than does the model, so themodel may overestimate the inflation factors for years following 2003.

The percentage growth rate represented by the exponential model is 4.21%. This can beconsidered the average percent increase in the inflation factors over these years. In other words,the average rate of inflation over this period was 4.21%.

Real Dollars versus Current Dollars

The CPI, and the inflation rates that represent the annual percentage increase in the CPI, keeptrack of how prices increase over time.91 The flip side of this is that money is worth less over time.If you had stuffed a dollar bill in the sofa in 2000, it would not be worth as much today becauseit could not buy as much as it could in the year 2000. To keep track of how the value of moneychanges due to inflation, and to adjust prices and other monetary amounts for inflation, we usewhat are known as constant dollars.

The terms current dollars or nominal dollars refer to an amount in actual dollars notadjusted for inflation. For example, suppose that in 1990, $1 would buy you 5 pieces of candy.You go into the store in 1990, lay down your dollar bill, and leave with 5 candies. The price ofone piece of candy in 1990 current dollars is 20 cents. Now, suppose you go back 13 years later, in2003, and you lay down your dollar bill and they only give you 2 candies. This means the price in2003 current dollars for one piece is 50 cents. The price of a piece of candy in current dollars, orthe nominal price, has gone up from 20 cents to 50 cents in 13 years. If you want to buy 5 candiesin 2003, you will have to pay $2.50.

Now, the 2 pieces of candy you got in 2003 for your dollar would have cost you only 40 centsin 1990. So, at least with respect to its power to purchase candy, your dollar in 2003 is only worth40 cents, compared to what the dollar was worth in 1990. The real value or purchasing power ofyour dollar has decreased from $1 to $.40 over that 13 year period (at least for this type of candy).

91It should be noted that prices sometimes decrease over time, and this is called deflation. Deflation can be takeninto account simply by letting the inflation factors decrease by the appropriate percentage. This in fact did happenduring the Great Depression, and at several other points in U.S. history

138

Constant dollars work the same way, except that instead of looking at just a piece of candy,we use the CPI, because it represents how prices of all consumer items have risen on average overtime. As with inflation factors, we pick a reference or base year. Then we find the value of a dollarin any other year in reference to that year.

For example, let’s say we pick 1980 to be our reference year. This means we are going tocompare the worth of a dollar in any other year to the worth of $1 in 1980. To find the value of adollar in any year N in constant dollars relative 1980, take the CPI in 1980 divided by the CPIfor the year N . For example, in Table 6 we see that the CPI for 1980 is 82.4. If we want the valueof a dollar in the year N = 2000 in real 1980 dollars, we take 82.4 divided by the CPI for 2000,which is 172.2. We get,

82.4172.2

= 0.478513357 ≈ 0.48

or 48 cents. So, a dollar in 2000 can only buy 48 cents worth of stuff compared to what a dollarbought in 1980.

Now, you may notice that to calculate the value of a dollar in real dollars, we are doing thesame thing we did for inflation factors except we are switching the numerator and denominator. IfR is our reference year, then

Inflation Factor for Year N with Reference Year R =CPI in Year N

CPI in Year R

Value of 1 Dollar in Year N in Real Dollars with Reference Year R =CPI in Year R

CPI in Year N

Here is another way to think about real versus constant dollars. If you had time travel capability,you could use it to make a lot of money by using the changing value of money. Suppose a particularitem sells for $1000 in 2000. If you travelled back from the year to 2000 to 1980, you could buythat same item for only $480 (48 cents on the dollar), assuming that the price of that item hadincreased at the same rate as inflation from 1980 to 2000. If you could bring it back to the year2000 and sell it, you’d make $520. You made money by spending it in 1980, when your money wasworth more.

Adjusting for Inflation

Of course, our hypothetical $1000 item may or may not have increased in price at exactly therate of inflation. The CPI, after all, is an average based on prices of many items, and some itemswill go up in price more than others. For example, you may have heard that health care costs (orcollege tuition) are rising faster than inflation. To adjust for inflation, simply take whatevernominal prices you are looking at and divide by the inflation factors. For example, Table 7 showsgasoline prices, based on national average, for a number of years starting in 1976. The nominalprices are shown in the second column, and inflation factors referenced to 1980 are in the thirdcolumn. The last column is the average price of gasoline in constant 1980 dollars. These valuesare found by taking the column two nominal prices divided by the column three inflation factors.

Notice that even though the nominal price in 2001 is more than twice the nominal price in1976, the real price, adjusted for inflation, is lower. We would say that gasoline prices have goneup more slowly than inflation from 1976 to 2001. Another way to say this is that gasoline priceshave fallen in real terms.92 The chart in Figure 5 shows both the nominal and the real pricesfor gasoline for all the years from 1976 through 2001. We have actually given the real prices twodifferent ways, once in 1980 dollars and once in 1996 dollars. Notice that the shapes of the graphsof the real prices are exactly the same! The only difference that changing the reference makes iswhere the plot of real prices crosses the plot of nominal prices. For the plot of real prices in 1980

92This is the same use of the word “real” made by Rush Limbaugh as quoted in Chapter Four


Nominal Inflation Price inYear Price Factors Real $ (1980)1976 0.61 0.691 0.881978 0.67 0.791 0.851980 1.25 1.000 1.251985 1.20 1.306 0.921990 1.16 1.586 0.731995 1.15 1.850 0.621997 1.23 1.948 0.631998 1.06 1.978 0.542000 1.51 2.090 0.722001 1.46 2.149 0.68

Table 7: Nominal and Real Gasoline Prices

dollars, the graph will meet the nominal graph in 1980, since that is where the nominal price equalsthe real price in 1980 dollars. Similarly, the nominal price will equal the real price in 1996 dollarsin the year 1996, and that is where these two plot intersect.

Wherever the graph of real prices is increasing, the nominal price is going up faster thaninflation. Wherever the real price is decreasing, the nominal price is going up more slowly thaninflation. Since the real prices in 2001 are lower than the real price in 1976, as noted above, nominalgasoline prices have gone up more slowly than inflation overall during that period.Unleaded Regular Gasoline Prices

00.511.522.51975 1985 1995 2005

NominalReal 1996 $Real 1980 $Figure 5: Nominal and Real Gasoline Prices

Based on the fact that real gasoline prices have been going down (and they were going downeven more steeply before the 1970’s), many politicians and others have said that complaints aboutexpensive gasoline are unwarranted. Americans actually spend much less for gasoline than almostany other people in the world, in real terms. If Americans in 2001 paid, in real terms, the sameamount for gasoline as in 1976, the nominal price of gasoline would have been $1.90 per gallon.

140

Conclusion

The above example illustrates one example of the use of exponential functions, as well as therelationship between exponential functions and any quantity that grows at a fixed percentagegrowth rate. Money in a savings account, the cost of college tuition, the size of a population,and the decay of a radioactive substance are all examples of situations which typically increase (ordecrease) by a fixed percentage. All of these can be modelled by exponential functions, just asquantities which grow at a constant rate can be modelled by linear functions.

Note that our base b, which is also referred to as the growth factor, for our exponential functionmodelling grandmother’s money was 1.005, while the monthly interest rate was 0.005. The 0.005is what we will call the percentage growth rate and it will be true in general that

The Growth Factor = 1 + The Percentage Growth Rate

This will be true regardless of what the percentage growth rate is, even if the percentage growthrate is negative (in which case we might call it the percentage rate of decay).

Reading Questions for Exponential Functions

1. For each of the following functions given in symbolic form, decide whether the functions islinear, exponential, power, or none of these.

(a) y = 13(100.015x)

(b) y = 3( 5√

x2)

(c) y = 2x2

(d) y = 4(x− 7) + 3(e) y = 6.23(1.016x)

(f) y = 23x0.75

2. How is a linear function similar to an exponential function? How is a linear function differentfrom an exponential function?

3. Each of the following is an exponential function. In each case, give the growth factor, they-intercept, and the percentage growth rate.

(a) f(x) = 4x

(b) f(x) = 3(13)x

(c) f(0) = 5, f(1) = 15(d) f(0) = 1, f(2) = 36

4. Let f(x) = abt be an exponential function which models the value of a particular car overtime as it ages. What do you know about the value of b? Justify your answer.

5. Give the equation for the exponential function whose y-intercept is 3.2 and whose growthfactor is 1.4.

6. Let f(x) = abx be the exponential function whose graph is given below.

? ?

-1

0

1

2

3

4

5

6

7

8

x -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

(a) What is the value of a? How do you know?(b) Is the value of b bigger or less than 1? How do you know?

141

142

7. Let h(x) = 5(4)x. Find each of the following.

(a) h(−3).(b) h(−1)(c) h(0)(d) h(8)

8. Determine if each of the following is an exponential function. If so, give its equation. If not,briefly explain why.

(a) 2y = 4x

(b)x −2 −1 0 1 2

f(x) 4 2 1 0.5 0.25

(c) The function describing the amount of money received by returning pop bottles if thedeposit is $0.10 per bottle.

9. Determine if each of the tables given below describes an exponential function. If so, give theequation for the function. If not, briefly explain why.

(a)x −2 −1 0 1 2

f(x) 1 3 9 27 81

(b)x −2 −1 0 1 2

f(x) 12

√2

2 1√

2 2

(c)x −2 −1 0 1 2

f(x) 0.08 0.4 0 0.4 0.08

(d)x −2 −1 0 1 2

f(x) 0.309 0.555 1 1.8 3.24

(e)x 0 2 4 6 8

f(x) 3 12 48 192 768

10. One of the following involves an exponential function and one involves a linear function.Identify which is which.

(a) Find the cost of gasoline if the price is $1.23 per gallon.(b) Find the population of a city that is increasing at a rate of 3.5% per year.

11. Let f(x) = xb. Explain why it is always true that the graph passes through the point (1, 1).

12. Evaluate each of the following:

(a) 161/2

(b) 47

(c) 1254/3

(d) 103/4

(e) e1.7

(f) 4e1.7


13. Write each of the following using radicals (i.e. write x1/2 as√

x).

(a) 3x9/5

(b) x4/3

(c) 6x54/5

(d) 8x7/4

14. Suppose you have $5000, and you want to save up to buy a new pickup truck to replace youraging 1998 Ford F-150. You estimate you can get at least 4 more years out of your currentvehicle, and so tentatively plan to invest the money for 4 years and use the balance as a downpayment on a new truck.

(a) Assume you can invest your money at fixed annual rate of 6% compounded quarterly.How much would you have at the end of 4 years?

(b) Instead of investing your money at fixed annual rate of 6% compounded quarterly, youdecide to take a somewhat riskier approach and invest in a variable rate account. Theupside is you can currently get an 8% annual rate in such an account. The downside isthe interest rate may go down, possibly significantly, before the 4 years is up.

i. If you are lucky, the interest rate will stay at 8%, compounded quarterly. How muchwould you have in the account after 4 years at this interest rate?

ii. If you are not lucky, the interest rate will go down. To keep things simple, supposethe account stays at 8% for the entire first year, goes to 6% during the second year,5% during the third, and 4% during the fourth.93 How much will you have after 4years under this scenario?

15. The table below gives child mortality data for Libya, Cuba, and the Bahamas.

Country 1990 1995 2000 2001Bahamas 29 23 17 16Cuba 13 10 9 9Mexico 46 36 30 29

(a) Find the exponential regression equation for the Bahamas’s child mortality rate. Itmight be convenient to let 0 represent the year 1990.

(b) Find the exponential regression equation for Cuba.(c) Find the exponential regression equation for Mexico.(d) Using your exponential regression equations, predict child mortality for each of these

countries for they years 2010 and 2050.

93In reality, interest rates in such accounts may vary from quarter to quarter.

144

16. The table below is similar to Table 5 in the text, except that (nominal) median householdincomes for the U.S. are given instead of the nominal price of gasoline. Fill in the householdincomes in real dollars in the last column. Have median incomes increased faster than inflationor not?

Nominal Inflation Income inYear Salary Factors Real $ (1980)1980 21,023 1.0001985 27,735 1.3061990 35,353 1.5861995 40,611 1.8501997 44,568 1.948

Table 8: Nominal Median Incomes and Inflation Factors, U.S.

17. Explain briefly the difference between real dollars and constant dollars.

18. Suppose the nominal price of a certain item increases at exactly the same rate as inflationfrom year to year. When the price is adjusted for inflation, giving the price in real dollars,what will be true about the price in real dollars from year to year?


Exponential Functions: Activities and Class Exercises

1. Moore’s Law.94

In 1965, Gordon Moore, cofounder of Intel Corporation, predicted that computing powerwould double roughly every 2 years or so.95 This prediction has come to be known as“Moore’s Law.”The data for Moore’s Law can be found in the Fathom file Moore’s Law Data. Uponopening the file, you should see a case table, two sliders, and a graph with a scatterplot ofthe data and an exponential model y = A · Bx. The dependent variable is the number oftransistors in an average CPU, a measure of computing power. The independent variable isthe year from 1970. The values of A and B are given by the sliders.

(a) Play with the sliders until the exponential model gives a reasonable approximation tothe data. What values of A and B did you use to give the best fit?

(b) Do these values seem to confirm Moore’s law? Explain carefully why or why not basedon the value of the growth factor B. You might want to find the value of B which wouldresult in a doubling every two years.

(c) As we will see in Chapter 8, a two-variable data set Y versus X will follow an exponentialmodel if Log(Y ) versus X follows a linear pattern. Add a new attribute Log Transistorsby calculating the logarithm of the Transistors variable as a formula.

i. Make a graph of Log Transistors versus Since1970. Is this a fairly linear graph?ii. Find the linear regression y = mx + b equation and the correlation coefficient for

Log Transistors versus Since1970.iii. As we will also see in Chapter 8, the exponential regression model for a data

set Y versus X has parameters A and B with A = 10b and B = 10m, where m andb are the slope and y-intercept of the linear regression equation for Log(Y ) versusX. Find these values of A and B for your regression equation.

iv. Use the sliders to change the values of A and B to those you found in the previouspart. How does this model fit the data? Does it seem to fit better than your originalmodel?

v. Finally, does this model seem to confirm Moore’s Law? Why or why not?

2. Stock Market Growth. The following table gives the Dow Jones Industrial Average (DJIA)on the last day of January for each year from 1970 through 2004.96

Year DJIA Year DJIA Year DJIA Year DJIA1970 800.4 1979 805 1988 1938.8 1997 6448.31971 838.9 1980 838.7 1989 2168.6 1998 7908.31972 890.2 1981 964 1990 2753.2 1999 9181.41973 1020 1982 875 1991 2633.7 2000 11497.11974 850.9 1983 1046.5 1992 3168.8 2001 10786.851975 616.2 1984 1258.6 1993 3301.1 2002 10021.561976 852.4 1985 1211.6 1994 3754.1 2003 8341.631977 1004.7 1986 1546.7 1995 3834.4 2004 10453.921978 831.2 1987 1896 1996 5117.1

94This activity is adapted from Fathom Dynamic Data Software: Workshop Guide, by Finzer and Erickson, p. 5495see Wikipedia at http://en.wikipedia.org/wiki/Moore’s law.96Data from www.djindexes.com

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(a) Graph a scatterplot of this data, letting x = 0 represent 1970. Which family of functions,linear or exponential, do you think is most appropriate to model this data? Explain.

(b) Find the exponential regression equation for this data.

(c) Graph your exponential regression equation along with the data. How well does theequation seem to fit the data?

(d) What percentage growth rate does your exponential equation represent? This percentagecan be considered the (effective) average annual growth rate in the DJIA.

(e) One type of investment that has become popular in recent years is an index fund. Anindex fund is a mutual fund which invests in the same stocks that are components of oneof the stock indexes. The idea is that the index funds performance should then mirrorthat of the index. If you bought a Dow Jones based index fund, your investment shouldgrow (and fall!!) along with the DJIA.

i. Using your average percentage growth rate from part (d), estimate what an invest-ment of $10,000 made in 1980 would be worth today.

ii. Assuming this average growth rate continues into the future, how much would$10,000 invested today in a Dow Jones Index Fund be worth 40 years from now?

(f) From the scatterplot, you can see that the DJIA has a fair amount of volatility. Inother words, although the general trend is upwards, there is a lot of up and down inthe graph. So, the average percentage growth rate is not a very good predictor of thegrowth rate in any particular year.

i. Create a new table which gives the actual percentage growth rate (or decline) in theDow for each year. You can do this by dividing the value in one year by the valuethe previous year, and then subtracting one.

ii. What is the largest percentage increase seen in these years?

iii. What is the largest percentage drop? How many years out of these 35 years did theDow experience a drop?

iv. Find the mean and the median of the percentage increases. How close are these tothe percentage growth rate represented by the exponential growth equation? Whatdo the mean and the median tell you about the skew of the distribution of thepercentage growth and decline rates?

v. (Optional bonus for those who follow the business news). Starting in 1996, the DJIAexperienced a large spike, peaking in 2000, and then declining until 2003. Can youthink of any explanation for this spike?

3. Comparing Car Models II. The data in the table below represent 38 car types of 7 differentmakes and 19 different models and variations for the 2005 model year. The data are fromwww.autos.msn.com and also appear in one of the activities in Chapter Five. As noted there,City Mileage, Highway Mileage, and Greenhouse Gas Emissions ratings are as determinedby the EPA. Gas emissions is given in tons of carbon dioxide per year under normal drivingconditions. Engine Size is in liters. Passenger Room and Luggage Room are given in cubicfeet.


City H’way Carbon Engine Passenger Lugg.Make Model Price mpg mpg Emiss. Size Room RoomHonda Civic 15325 29 38 5.9 1.7 91 10Honda Accord (2dr) 22622.5 24 34 7 2.4 103 14Honda Accord (4dr) 22622.5 24 34 7 2.4 103 14Toyota Avalon 23500 22 31 7.6 2.2 107 14Honda Accord (2dr) 22622.5 21 30 7.8 3 103 14Honda Accord (4dr) 22622.5 21 30 7.8 3 103 14Toyota Corolla 15667.5 30 38 5.7 1.8 84 14Honda Civic 15325 29 38 5.9 1.7 91 10Ford Crown Victoria 27475 18 25 9.2 4.6 111 21Toyota Matrix 4x2 16855 28 34 6.2 1.8 94 22Toyota Matrix 4x4 16855 28 34 6.2 1.8 94 22Toyota Celica 18600 29 36 6 1.8 78 17Toyota Camry 21875 24 34 6.9 2.4 102 17Chevrolet Cobalt (4dr) 17527.5 24 32 6.7 2.2 87 14Ford Taurus SW 13875 19 25 8.8 3 109 39Toyota Camry 21875 21 29 8 3.3 102 17Ford Taurus 22165 20 27 8.4 3 105 17Toyota Camry 21875 20 28 8.2 3 102 17Chevrolet Monte Carlo 25317.5 21 32 7.6 3.4 96 16Toyota Corolla 15667.5 30 38 5.7 1.8 84 14Chevrolet Aveo 11160 26 34 6.5 1.6 91 12Saturn Ion (2dr) 11900 24 32 6.7 2.2 91 15Saturn Ion (4dr) 11900 24 32 6.7 2.2 91 15Ford Focus 15842.5 26 32 6.8 2 94 15Chevrolet Malibu Classic 21630 25 34 6.7 2.2 99 16Toyota Camry 21875 24 34 6.9 2.4 102 17Honda Accord (4r) 22622.5 24 34 7 2.4 103 14Chevrolet Cobalt (4dr) 17527.5 24 32 6.7 2.2 87 14Dodge Stratus (4dr) 21845 22 30 7.6 2.4 94 16Toyota Camry 21875 21 29 8 3.3 102 17Honda Accord (4r) 22622.5 21 30 7.8 3 103 14Dodge Stratus (4dr) 21845 21 28 8.1 2.7 94 16Toyota Camry 21875 20 28 8.2 3 102 17Dodge Stratus (2dr) 21845 21 28 8 2.4 86 16Dodge Stratus (2dr) 21845 20 28 8.3 3 86 16Dodge Neon 13750 25 32 6.8 2 90 13Chevrolet Cavalier (2dr) 14017.5 24 34 6.7 2.2 92 14Chevrolet Cavalier (4dr) 14017.5 24 34 6.7 2.2 92 14

(a) Make a scatterplot of greenhouse gas emissions versus city mileage, and graph the linearregression line along with the scatterplot.

i. Notice that data points at the left and right end of the scatterplot tend to be abovethe line and those in the middle tend to be below the line. This suggests that adecreasing exponential function might provide a better fit to the data than a linearfunction. Find the exponential regression equation for this data and graph it alongwith the scatterplot.

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ii. Which do you think provides a better fit, the linear regression or the exponentialregression equation. Recall that, as discussed in the text, it is not valid to use thecorrelation coefficients to answer this question.

iii. Predict the greenhouse gas emissions for a car that gets 40 miles per gallon in thecity using both the linear and the exponential models. How much difference is therein these predictions?

iv. Repeat the previous part for a car that gets 10 miles per gallon in the city.

(b) Make a scatterplot of greenhouse gas emissions versus average price in thousands.

i. Looking at the scatterplot, which family of functions do you think is more likely toproduce a better fit to these data, linear or exponential?

ii. Find the exponential regression model for these data.iii. What is the growth factor and percentage growth rate for your exponential model?

What does the percentage growth rate mean in terms of emissions and price?iv. What does your exponential model predict for a car that costs $50,000?

4. Airline Deregulation. The graph below is from a report by the Heritage Foundation onairline deregulation. Congress passed deregulation legislation in 1978, cutting back drasticallyon price, route, and other restrictions that airline companies had been required to followbefore then.

The foundation supported deregulation of the airline industry, and the report tries to makethe case that, based on 20 years of experience, deregulation has been good both for theindustry as a whole and for consumers. The foundation cited this chart to support the claimthat, after deregulation, the rate of increase in miles flown has been greater than the rate ofincrease prior to deregulation. This claim was meant to respond to critics of deregulation whohave claimed that deregulation led to fewer options and a lower rate of service for consumersin many areas of the country. Here is a short excerpt from the report.97

Airlines also are logging more miles than before deregulation. Whereas carriersflew roughly 2.5 billion miles in 1978, they logged more than double that numberlast year alone, flying approximately 5.7 billion miles in 1997. Furthermore, asChart 5 illustrates, this trend increased much more rapidly after deregulation thanit did before market liberalization.

(a) The bars represent the actual number of miles flown by U.S. domestic air carriers. Whatfunction family do you think would be most appropriate to model this data?

(b) The black line, up to the year 1978, is meant to represent what is called the three-yearmoving average. This is typically calculated by taking the values of three consecutiveyears, including the year in question, and averaging them. What seems to be wrongwith the three-year averages as plotted here?

(c) In the excerpt from the report given above, it was noted that the trend of miles flownincreased much more rapidly after deregulation than before. The implication is thatmore miles were flown after deregulation than would have been flown if deregulationhad not occurred.

i. What function family is being used in creating the extrapolated trend line?ii. In order to create the extrapolated line, the author probably calculated the linear

trend line for years up to 1978. Why do you think the author would do this? Whatis wrong with using a linear model in this situation?

97The full report can be found at http://www.heritage.org/Research/Regulation/BG1173.cfm. The author isAdam D. Thierer, former Alex C. Walker Fellow in Economic Policy at The Heritage Foundation, and now at theCato Institute. Mr. Thierer has graciously allowed the use of his report for this activity.


Figure 6: Billions of Aircraft Miles

iii. The actual data on which the graph is based were given later in the report. Thenumber of miles flown in 1950 was about .48 billion, and the number flown in 1970was about 2.39 billion. Letting t = 0 represent 1950, use these two points to createan exponential model M = a(bt) where M represents estimated number of milesflown and t represents time in years from 1950.

iv. Use your exponential model to estimate the number of miles flown in 1980, 1990,and 1996. How do these values compare with those indicated by the graph?98

v. One might argue that the years 1950 and 1970 used to create the previous modelwould lead to an overestimate, since the graph has a peak near 1970. Picking alower point from a later year would give a different model with lower predictionsfor future years. In the year 1974, only 2.18 billion miles were flown. Use thispoint together with 1950 to create an exponential model, and then use this modelto predict the number of miles flown in 1980, 1990, and 1996.

vi. What are the percentage increases in numbers of miles flown represented by yourtwo exponential models?

(d) Comparing your two exponential models to the actual data, and using the other infor-mation you have discovered, what would you say about the Heritage Foundation’s claimthat the rate of increase in the number of miles flown was higher after deregulation thanit was before deregulation?

5. Inflation and investing. We saw in the text that one effect of inflation is that money isworth less over time. This includes not only cash in your wallet, but also other monetaryamounts like the value of your stock portfolio.The table in activity 2 gives data on the Dow Jones Industrial Average (DJIA) over time,starting in 1970. Although the DJIA is not measured in dollars, we could consider the

98The actual values in these years were 2.82, 4.49, and 5.5 billion miles

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amounts as representing the value of an investment as it grows over time. Thus, we couldsay that an investment valued at $800.40 in 1970 would grow to $10453.92 in 2004 if it grewat the same rate as the DJIA.However, these values would be in nominal dollars. Inflation would make the value of theinvestment grow more slowly in real terms.

(a) Using Table 6 in the text, which gives the CPI and inflation factors every five yearsstarting in 1935, create a table showing the value of an investment that mimics theDJIA, but adjusted for inflation to real 1970 dollars. Since you only have inflationfactors for every fifth year, you need only adjust the DJIA values for those same years.

(b) Using your table from the previous part of this activity, find an exponential model forthe DJIA adjusted for inflation, starting in 1970.

i. What is the percentage growth rate of the DJIA in real terms?ii. If you invested $1000 today and it grew exponentially according to this percentage

growth rate, what would it be worth in 40 years in real dollars, indexed to today?

6. Population Growth. Each of you should collect data on the population of a particularcountry. You should find yearly data going back to at least 1970. For your country, answerthe following questions.

(a) Find the exponential model that best fits your data. Then discuss how well the expo-nential model fits the data. Are there periods of time for which the model fits better orworse than at other times?

(b) What is the average percentage growth rate of the population over the period you’vemodeled? (This is the percentage growth rate represented by your exponential model)

(c) Use your exponential model to predict the population for your country 5 years from nowand 10 years from now.

(d) Find each individual year’s percentage growth rate. To find the percentage growth rateof the population in a given year N, take the year N population and divide it into thepopulation for the next year N+1, and then subtract 1. (You can do this for all yearsat once on a calculator or on a spreadsheet).

(e) Graph the percentage growth rates you found in part (d) over time. How are theychanging over time? How have they been changing over the last 10 years? Select afamily of functions to model the percentage growth rate over the last several years(anywhere from 5 to 20, depending on how “nice” the data seems to be), and find themember of this family which best fits the data. Discuss how well you think your functionfits the data.

(f) Use the function you found in part (e) to project what the percentage growth rate shouldbe for each of the next 10 years. Give the results in a table.

(g) Now, use the results from part (e) to make new predictions for the population of yourcountry 5 years from now and 10 years from now. Compare these to the predictionsyou made in part (c) Discuss the differences between the two sets of predictions. Alsodecide which set of predictions you have more confidence in and why.

7. Logarithmic Functions

So far, we have looked in detail at linear and exponential functions. While different in manyways, these functions are similar in that, for the most part, they only involve the basic arithmeticoperations of addition and multiplication.99 Logarithmic functions, while they “look” different anddo not explicitly involve basic arithmetic operations, are actually closely related to exponentialfunctions. In fact, the most succinct way to describe logarithms is that:

Logarithms are exponents.

Let’s see exactly what this means.

Definitions

Any exponential function y = abx involves two parameters. The parameter b is called the base,and the exponent is the independent variable. Logarithmic functions will also involve a parameterb called the base, and an exponent as one of the variables, but in this case the exponent will bethe dependent variable. For any base b, we define the logarithm base b as follows.

y = logb x means that by = x

In other words, logb x represents the exponent y such that by = x. This is what we mean whenwe say “logarithms are exponents.” The first equation above is called the logarithmic form andthe second exponential form.

Example 1. Using the definition above, find the following logarithms.

1. log3 9

2. log3 1

3. log2 8

4. log2(1/4)

Solution:

1. log3 9 is the exponent y such that 3y = 9. Since 3y = 9 = 32 and the function f(x) = 3x is aone-to-one function, we must have that y = 2. Therefore, we have log3 9 = 2.

2. We are looking for the exponent y such that 3y = 1. Since any nonzero number to the 0power is 1, we have log3 1 = 0. In fact, for any positive base b, logb 1 = 0, since b0 = 1.

3. Since 23 = 8, log2 8 = 3.

4. We are looking for the exponent y so that 2y = 14 . Recalling that negative exponents give

reciprocals, we can deduce that −2 is the correct exponent here. 2−2 = 1/22 = 1/4. So,log2(1/4) = −2.

Now, what if someone asks you to calculate log2 10? They want to know the exponent y suchthat 2y = 10. This is not as easy as the examples above! We know that 23 = 8 and 24 = 16, so wecan deduce that this exponent is somewhere between 3 and 4. We could start guessing at numbers

99Exponents are really just repeated multiplication. To be fair, exponents that are not integers are a bit morecomplicated, but in this course, we will not consider the theoretical mathematical underpinnings of non-integralexponents.

151

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for y, and then calculating 2y until we find one that gets us close to 10, but this does not soundlike much fun. Fortunately, technology will come to our aid.

Your calculator very likely has two buttons to calculate logarithms. The one which we willtypically use is the LOG button. The LOG button calculates logarithms with base 10. Forexample, if you press LOG 100 your calculator should give you a result of 2, which makes sensesince 102 = 100. If you press LOG 35 you should get 1.54407, or something close to this, dependingon how many digits your calculator displays. If you check, you will see that 101.54407 gives you ananswer pretty close to 35. It will not be exact, since we did some rounding off. If you see a logwith no base given, this will typically mean the logarithm base 10.

The second logarithm button on your calculator is probably labelled LN. This calculates loga-rithms with base e, where e ≈ 2.718281828. The logarithm base e is called the natural logarithm,and is used extensively in scientific fields. The appendix contains more background on the naturallogarithm, and where the number e comes from.

What about our problem with log2 10? We can use the following property of logarithms tocalculate this.100

logb x =log x

log b

Let’s apply this property to our example. We get,

log2 10 =log 10log 2

≈ 10.30103

≈ 3.322

To check, calculate 23.322 on your calculator.

The Graph of the Logarithm Function

Although there is a whole family of logarithmic functions y = logb x, one for each b, we willpredominantly use y = log x, the base 10 logarithm, or lnx the logarithm base e.101 Becauselogarithms are exponents, we will be able to relate the characteristics of the function y = log x tothe function y = 10x. Table 1 shows some values for both of these functions. 102

Table 1: The logarithm function and the exponential function base 10.

x 0.01 0.1 .5 1 2 5 10 20 30 40 50 100log x −2 −1 -0.30 0 0.30 0.70 1 1.30 1.477 1.602 1.699 2

x −2 −1 -0.30 0 0.30 0.70 1 1.30 1.477 1.602 1.699 210x 0.01 0.1 .5 1 2 5 10 20 30 40 50 100

Notice that the tables are exactly the same except that the dependent and independent variablesare reversed!! In other words, if a point (u, v) is in the table (and thus on the graph) for y = log x,then the point (v, u) is in the table (and thus on the graph) for y = bx. The effect of this is thatthe graph of y = log x will have the same shape as the graph of y = 10x, but with the x and yaxes switched. In other words, if you have the graph of y = 10x on a transparency, flipping thetransparency over diagonally so that it is now face down and with the x-axis running verticallywill give you a picture of the graph of y = log x.

Two versions of the graph are shown in Figure 1. On the left, the window is 0 ≤ x ≤ 110,−1 ≤ y ≤ 2.5. In this window, we see the very slow growth of the logarithmic function as xincreases, and also the vertical asymptote at x = 0. The version on the right is an extreme close-up

100Proofs for this and the other properties of logarithms we give later are found in the Appendices.101As you can see in the appendices, b must be positive and not equal to 1.102We have rounded many of the values, but they are all accurate to within a hundredth.

7. Logarithmic Functions 153

of a section of the graph very close to the y-axis (notice the horizontal scale). This section of thegraph would be indistinguishable from the y-axis in any window including x values 2 or larger, buthere we see that the graph is indeed not touching the y-axis.

y = x( )log

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

x0 20 40 60 80 100

no data Function Plot

y = x( )log

-3.0

-2.8

-2.6

-2.4

-2.2

-2.0

-0.004 0.000 0.004 0.008 0.012x


Figure 1: Two views of the logarithmic function base 10.

Since the graph of the logarithmic function y = log x has the same shape as the exponentialfunction y = 10x except that it is “flipped diagonally,” we have the following comparisons betweenthe two graphs, and these hold for logarithmic and exponential functions of bases b other than 10as long as b > 1.

Exponential Functions Logarithmic FunctionsAlways concave up Always concave down

y-values always positive x-values always positivey-intercept equal to 1 x-intercept equal to 1

Horizontal asymptote y = 0 Vertical asymptote x = 0as x goes to negative infinity as y goes to negative infinity

Graph is very steep for large x Graph is very flat for large x.

Finally, recall that for the exponential function with base b, the result of increasing the inputby 1 is that the output gets multiplied by b. For y = log x, this means that if we multiply theinput by 10 (the base b), the output will increase by 1. You can see this in the table, for example,by comparing the outputs for log x for the inputs 5 and 50, or for the inputs 2 and 20, or for theinputs 0.1, 1, 10, and 100. We could write this property in equation form as

log 10x = 1 + log x.

Properties of Logarithms

To summarize what we know so far about logarithms, we have the following properties of thelogarithm function.

• log 10x = x.

• log(1) = 0.

• log x is negative for 0 < x < 1 and positive for x > 1.

• y = log x has a domain of x > 0 and a range of all real numbers.

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• The graph of y = log x is increasing and concave down.

• logb x = log xlog b for any positive b 6= 1.

There are four more properties that we will also find useful. These are:

1. log(uv) = log u + log v.

2. log(u/v) = log u− log v.

3. log xn = n log x.

4. 10log x = x.

The first property is a generalization of the equation log 10x = 1+log x given above. Note thatlog 10 = 1. So,

log 10x = log 10 + log x = 1 + log x

The more general property 1 says we can substitute any other number u for the 10 in log 10xby using log u instead of log 10. Why does this property work in general?

Let’s illustrate with an example. Let a and b be numbers such that log(a) = 3 and log(b) = 4.This means a = 103 and b = 104. Notice that log(a) + log(b) = 3 + 4 = 7. What input would givean output of 7 under the log function? It would be 107, the product of the original two inputs 103

and 104, since log 107 = 7. Since ab = 107, we get

log ab = log 107 = 7 = 3 + 4 = log a + log b

In general, property 1 is a consequence of the fact that, whenever you multiply two numberswith exponents that have the same base, you add the exponents. Symbolically, if log a = p andlog b = q, then 10p = a and 10q = b. So ab = 10p10q = 10p+q (we’re adding the exponents). Thismeans log(ab) = p + q = log(a) + log(b).

Discussion of why properties 2 and 3 work are given in the appendices.Let’s consider property 4. Suppose we start with 10log x = y. If we change this equation to

logarithmic form, we get

log10 y = log x

or just log x = log y. Since the logarithm function is increasing, the only way this can happen is ifx = y. Combining this with our original equation 10log x = y, we get 10log x = x.

In interpreting these properties, it is important to keep in mind how logarithms fit into the usualorder of operations. In general, unless there are parentheses involved, logarithms are done afterany multiplications, exponents, or divisions that come immediately after the log sign, but beforeany additions or subtractions and before any multiplications immediately preceding the log sign.Refer to the appendices as needed for more examples and discussions on this and other propertiesof logarithms.

Why are these properties important? One reason is that knowing these properties gives youa better understanding of how logarithms work. A second is that they will be useful at times insolving problems that involve either logarithms or exponential functions.

The Natural Logarithm

So far, we have concentrated on the logarithm base 10. The other commonly used logarithmis the logarithm base e, also referred to as the natural logarithm, and denoted in symbols as ln.Recall e is an irrational number approximately equal to 2.71828. By definition

y = lnx means that ey = x.


Example 2. Using the definition above, find the following logarithms.

1. ln e2

2. ln 1

3. ln(1/e)

4. ln 10

Solution:

1. ln e2 is the exponent y such that ey = e2. The only possible solution for y is 2. So ln e2 = 2.In fact, the same logic implies that ln ex = x for any x.

2. We are looking for the exponent y such that ey = 1. Since any nonzero number to the 0power is 1, we have ln 1 = 0.

3. Since 1e = e−1, we get ln(1/e) = ln e−1 = −1.

4. We are looking for the exponent y so that ey = 10. Unless we want to use the “guess andcheck” method, we would evaluate ln 10 using a calculator. We get ln 10 ≈ 2.3026. To check,we can evaluate e2/3026 to get 10.00015 or very close to 10.

Properties of the Natural Logarithm

Previously, we gave a number of properties satisfied by the common base 10 logarithm. Thenatural logarithm satisfies essentially these same properties, as do all logarithms, and for the samereasons. Explicitly, we have the following properties of the natural logarithm.

• ln ex = x.

• ln(1) = 0.

• lnx is negative for 0 < x < 1 and positive for x > 1.

• y = lnx has a domain of x > 0 and a range of all real numbers.

• The graph of y = lnx is increasing and concave down.

1. ln(uv) = lnu + ln v.

2. ln(u/v) = lnu− ln v.

3. lnxn = n ln x.

4. eln x = x.

Example 3.Use the properties of the natural logarithm to express the following expressions using only lnx,

ln y, and ln z.

1. lnx4y

2. ln√

xyz

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Solution:

1. We have

lnx4y = lnx4 + ln y by property 1= 4 lnx + ln y by property 3

2. First, recall that x1/2 =√

x. We have,

ln√

xyz = ln

√x− ln yz by property 2

= lnx1/2 − (ln y + ln z) by property 1= 1

2 lnx− ln y − ln z by property 3 and the distributive law

Logarithmic Regression

In Chapter Six, we used exponential regression as a standard way to find a “best fit” exponentialmodel for a given set of data. We would tend to apply exponential regression to data where thegraph is increasing (or decreasing) and concave up.

Given the basic shape of the graph of the logarithmic function, we would consider using alogarithmic model when the graph is increasing and concave down, and contains only positive xvalues. As with exponential regression, a number of software packages and graphing calculatorshave logarithmic regression features. Typically, these give equations of the form y = a + b ln x ory = a + b log x.

Example 4. Table 2 shows how life expectancies changed in the United States during the 20thcentury.103 The value given for each year represents how long a person born in that year couldexpect to live, on the average. Find a logarithmic model for this data, and discuss how good thefit is. Use the model to predict the life expectancy for U.S. children born in 2010 and 2050. Howreasonable do these predictions seem to be?

Year 1900 1915 1930 1945 1960 1975 1990 2000Life Expectancy 47.3 54.5 59.7 65.9 69.7 72.6 75.4 76.4

Table 2: U.S. Life Expectancies

Solution: Figure 2 shows a scatterplot of the data. The pattern seems to be slightly concavedown. This indicates that a logarithmic function might be an appropriate model.

If we let the independent variable be years from 1900, however, we will have to take into accountthat we cannot use 0 as a value when taking logarithms (since log(0) is not defined). To adjustfor this, we leave off the first data point and construct the plots using only the years starting with1915, which corresponds to an x-value of 15.

Using logarithmic regression on a TI calculator, we get the logarithmic model is y = 12.05 lnx+20.521, which uses the natural logarithm. The original data with the model is graphed in Figure3. It seems to fit the data well.

If you prefer, you can change the lnx in the model to log x using the change of base formula.

ln x =log x

log e≈ log x

.434494≈ 2.3026 log x

103This example is adapted from Functioning in the Real World by Sheldon Gordon, et. al., p. 189

7. Logarithmic Functions 157U.S. Life Expectancy from 1900404550556065707580

0 15 30 45 60 75 90 105Figure 2: U.S. Life Expectancy

This changes the form of the equation to y = 12.05(2.3036 log x) + 20.521 or y = 27.746 log x +20.521. This is really the same equation, and will give the same estimates as the equation usingthe natural logarithm lnx.

U.S. Life Expectancy with Logarithmic Model

0102030405060708090

0 25 50 75 100

Figure 3:

For the year 2010, the logarithmic model predicts a life expectancy 77.02.

Solving Exponential Equations

One very important use of logarithms is in solving exponential equations. An exponentialequation is any equation where the variable appears as an exponent. Let’s begin with an example.

More on Compound Interest

In Chapter Six, we considered a scenario in which your grandmother invested $2,000 when you

158

were born in order to help you with your college tuition. At a 6% annual interest rate compoundedmonthly for 18 years you had a total of $5873.53 to apply towards your first year.

Suppose, however, that you have been successful in obtaining your tuition dollars by othermeans and don’t need grandmother’s money now. You decide to keep saving the money until youhave $10,000, at which point you intend to use it as a down payment on a house or perhaps for anew car. How long will it take until you have your $10,000? To answer this question, we will uselogarithms, principally property 3 in the list of properties of logarithms.

In this situation, we want to know for which value of t will B(t) be equal to $10,000, whereB(t) = P (1 + i)t. We can set this up as an equation, using P = $2000 and i = 0.06/12 = 0.005.Since we are compounding monthly, t represents the number of months. We obtain the equation

10, 000 = 2000(1.005)t.

We want to solve this for t. In general, whenever you want to solve an equation where thevariable is an exponent, use logarithms. First, however, we divide both sides by 2000 to get

5 = (1.005)t

Then we take the logarithm of both sides of the equation, and use property 3.

log 5 = log(1.005)t = t · log 1.005

t =log 5

log 1.005≈ 322.7

You will have your $10,000 in 323 months (from your birth) which means you can have yourhouse or your car before your 27th birthday.

This example illustrates the general use of logarithms in solving exponential equations. Hereare a few more examples.

Example 5.Solve the exponential equations.

• 12x = 456

• 4(5x+1) = 76

Solution:

• We start solving the equation 12x = 456 by taking the logarithm of both sides. We can useany base, but base 10 and base e are the most used.

log 12x = log 456

x log 12 = log 456

x =log 456log 12

≈ 2.6591.079

≈ 2.464

We can check this by computing 122.464.


• To solve 4(5x+1) = 76 we first divide by 4.

4(5x+1) = 76

4(5x+1)4

=764

= 19

log 5x+1 = log 19

(x + 1) log 5 = log 19

x + 1 =log 19log 5

x =log 19log 5

− 1 ≈ .8295

Reading Questions for Logarithmic Functions

1. Express the following without using logarithms. You should not need a calculator.

(a) log2 32(b) loga a2

(c) log 100, 000, 000(d) logb b

(e) log218

(f) ln e

(g) ln e6

(h) ln ex+1

2. Solve the following equations.

(a) 3x = 81(b) 3x = 110(c) 2 · 3x = 110(d) 3 · 2x = 110(e) 5 · 1.3x+2 = 294(f) 2365 = 12(1.016)x

(g) 2x−3

5 = 11.7

3. Suppose you have $5000, and you want to save up to buy a new pickup truck to replace youraging 1998 Ford F-150. You estimate you can get at least 4 more years out of your currentvehicle, and so tentatively plan to invest the money for 4 years and use the balance as a downpayment on a new truck.

(a) Assume you can invest your money at fixed annual rate of 6% compounded quarterly.How much would you have at the end of 4 years?

(b) As in part a), assume you can get at annual rate of 6% compounded quarterly. However,at the end of the 4 years, you decide your current truck is doing fine, and so decide towait to buy the new truck until you have $8000 in the account. How much longer untilthis happens?

(c) Instead of investing your money at fixed annual rate of 6% compounded quarterly, youdecide to take a somewhat riskier approach and invest in a variable rate account. Theupside is you can currently get an 8% annual rate in such an account. The downside isthe interest rate may go down, possibly significantly, before the 4 years is up.

i. If you are lucky, the interest rate will stay at 8%, compounded quarterly. How longwould it take for this account to grow to the same balance as you had in part a),where you earned 6% and left the money in the account for 4 years?

ii. If you are not lucky, the interest rate will go down. To keep things simple, supposethe account stays at 8% for the entire first year, goes to 6% during the second year,5% during the third, and 4% during the fourth.104 How much will you have after 4years under this scenario?

104In reality, interest rates in such accounts may vary from quarter to quarter

160


4. Match each of the following functions in symbolic form with the appropriate graph.

(a) y = 3x

(b) y = log2 x

(c) y = 10 log2 x

(d) y = log8 x

(e) y = 2 · 1.5x

(f) y = log2 4x

-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10

-4

-3

-2

-1

0

1

2

3

4

0 2 4 6 8 10

0

50

100

150

200

250

300

0 1 2 3 4 5 -40

-30

-20

-10

0

10

20

30

40

0 2 4 6 8 10

0

5

10

15

20

25

0 2 4 6 8 10

0

2

4

6

8

10

12

14

16

0 1 2 3 4 5

5. Solve the following equations. Changing to exponential form may help.

(a) log3 x = 4(b) logx 100 = 2(c) lnx = 2.5

162

6. For each step, say which of the properties 1 through 4 of the natural logarithm are beingused.

ln a · bx = ln a + ln bx

= ln a + x ln b

7. For each step, say which of the properties 1 through 4 of the natural logarithm are beingused.

ln a · xb = ln a + ln xb

= ln a + b ln x

8. Use properties 1 through 4 of the natural logarithm to write the given expression in so thatany logarithms are only lnx, ln y, or ln z.

(a) lnxy2

(b) lnx2y3

(c) ln xz

(d) ln zx2

y

(e) ln√

xyz

9. What does it mean when we say “logarithms are exponents?”

10. Say whether the following are true or false.

(a) ln(1) = 0.(b) ln(uv) = lnu · ln v.(c) ln ex = e.(d) The graph of y = lnx is increasing and concave down.(e) ln(u/v) = lnu− ln v.(f) lnxn = n lnx.

(g) eln x = x.(h) y = lnx has a domain of x > 0.


Logarithmic Functions: Activities and Class Exercises

1. Airline Revenues.

Table 3: Airline Yields for Domestic U.S. Flights

Year Nominal Real (78) Log of Year Nominal Real Log ofYield Yield(78$) Real Yield Yield Yield(78$) Real Yield

1926 12.03 44.31 1.647 1964 6.12 12.87 1.1101927 10.6 39.72 1.599 1965 6.06 12.54 1.0981928 11 41.94 1.623 1966 5.83 11.73 1.0691929 12 45.75 1.660 1967 5.64 11.01 1.0421930 8.3 32.4 1.511 1968 5.61 10.51 1.0221931 6.7 28.74 1.458 1969 5.79 10.29 1.0121932 6.1 29.03 1.463 1970 6 10.08 1.0031933 6.1 30.59 1.486 1971 6.33 10.19 1.0081934 5.9 28.71 1.458 1972 6.4 9.98 0.9991935 5.7 27.13 1.433 1973 6.63 9.74 0.9891936 5.7 26.74 1.427 1974 7.52 9.95 0.9981937 5.6 25.36 1.404 1975 7.69 9.32 0.9691938 5.18 23.45 1.370 1976 8.16 9.35 0.9711939 5.1 23.92 1.379 1977 8.61 9.26 0.9671940 5.07 23.61 1.373 1978 8.49 8.49 0.9291941 5.04 22.35 1.349 1979 8.96 8.05 0.9061942 5.34 21.36 1.330 1980 11.49 9.09 0.9591943 5.35 20.16 1.304 1981 12.74 9.14 0.9611944 5.34 19.78 1.296 1982 12.02 8.12 0.9101945 4.95 17.93 1.254 1983 12.05 7.89 0.8971946 4.63 15.48 1.190 1984 12.8 8.03 0.9051947 5.05 14.77 1.169 1985 12.21 7.4 0.8691948 5.76 15.58 1.193 1986 11.08 6.59 0.8191949 5.78 15.83 1.199 1987 11.45 6.57 0.8181950 5.56 15.04 1.177 1988 12.31 6.78 0.8311951 5.61 14.07 1.148 1989 13.08 6.88 0.8381952 5.57 13.7 1.137 1990 13.43 6.7 0.8261953 5.46 13.33 1.125 1991 13.24 6.34 0.8021954 5.41 13.11 1.118 1992 12.85 5.97 0.7761955 5.36 13.04 1.115 1993 13.74 6.2 0.7921956 5.33 12.78 1.107 1994 13.12 5.77 0.7611957 5.31 12.32 1.091 1995 13.52 5.78 0.7621958 5.64 12.72 1.104 1996 13.76 5.72 0.7571959 5.88 13.17 1.120 1997 13.97 5.68 0.7541960 6.09 13.41 1.127 1998 14.08 5.63 0.7511961 6.28 13.69 1.136 1999 13.96 5.46 0.7371962 6.45 13.93 1.144 2000 14.57 5.52 0.7421963 6.17 13.15 1.119 2001 13.25 4.88 0.688

2002 11.98 4.34 0.637

164

December, 2003 marked the 100th anniversary of the world’s first powered flight, conductedby the now famous Orville and Wilbur Wright. Since then, air travel has mushroomed into ahuge industry with revenues of over 100 billion dollars annually. This activity considers howairline revenues have changed over the first century of flight.One standard measure of airline revenues is called the yield. In short, the yield is the amountof revenue taken in per person per mile flown. For a single flight, this can be calculated bytaking the total revenue from all paying passengers, dividing by the number of passengers(giving the average fare per passenger), and then dividing this by the number of miles. This istypically converted to cents per person per mile. The accompanying graph shows the averageairline (Nominal) yields for all U.S. domestic flights for the years 1926-2002, along with realyields. The real yields are the nominal yields adjusted for inflation. In this chart, real yieldsare calculated in 1978 real dollars. The table of the data is given above.Airfare Yields in Cents Per Person Per Mile - Domestic

051015202530354045501920 1940 1960 1980 2000

Nominal YieldReal Yield (78 $)Figure 4: Graph of Airline Yields - Nominal and Adjusted for Inflation

(a) Using nominal yields, about how much did it cost a person to fly a mile in 1940? Abouthow much did it cost in 2000?

(b) The nominal yields decreased sharply during the early 1930’s. Why do you think thishappened? You could discuss general sorts of reasons which might occur during anyperiod, or particular historical events that were occurring during that period.

(c) During the 1970’s, nominal yields increased sharply, and yet real yields continued todecline at roughly the same rate. Explain how this is possible using the idea of inflation.

(d) The graph of real yields decreases steadily throughout this period, and looks as if it mightfollow an exponential curve with a negative percentage growth rate. Use exponentialregression to find an exponential model for real yields as a function of time. Instead ofusing all of the data, you could use data from just the years which are multiples of five(1930, 1935, etc.). Graph a scatter plot of the data you used along with your regressionequation. Visually, how well does the curve seem to fit the data?

(e) We have seen that, when using linear regression, it is possible that the correlationcoefficient may imply a fairly strong correlation, and yet the scatter plot clearly shows alinear model is not appropriate. The same thing might occur with exponential regression,as discussed in the text, and we can test visually whether an exponential model isappropriate by taking the Log of the dependent variable data, and seeing whether this


results in a linear model. Take the Log of the yield data you used in part (d), and thencreate a scatter-plot of the Log of the real yields versus time. Does this relationshipappear to be linear? If so, estimate the slope of a line that would model this data (youmay do this from the graph or use linear regression on the Log of real yields data). Ifm is the slope, calculate 10 to the m. How close is this to the growth factor for yourregression equation?

(f) The graph of nominal yields seems to have taken a sharp decrease after the year 2000.Why do you think this happened?

(g) How can you tell from the table that the real yields are calculated using 1978 dollars?

2. Flood Damages. In both chapter four and chapter six, we considered data on U.S. FloodDamages. The scatterplot from chapter six is recreated below along with the exponentialmodel given by the formula D1(t) = 0.741(1.0168t). Here, t gives the year from 1900, andD(t) is total flood damages in billions of dollars. Note that the equation looks different thanthe one given in the graph, but is essentially the same as e0.0167 ≈ 1.0168.

US Flood Damages Adjusted for Inflationy = 0.7414e0.0167x

$0.00

$4.00

$8.00

$12.00

$16.00

$20.00

0 20 40 60 80 100

Year from 1900

Dam

ages

in B

illio

ns

of

$

Figure 5: Two functional representations of US Flood Damages.

(a) What is the percentage growth rate represented by the exponential model D1?(b) What does the exponential model predict for the years 2050 and 2100?(c) According to the model, when will flood damages reach 25 billion, and also 50 billion

dollars?(d) The data shows a lot of variability, and so the exponential model is likely to give esti-

mates very different than the actual damages, both in past and future years. Create anadditional model which will give estimated upper limits on flood damages. To createthe “upper limit model,” you could estimate the values for several of the “ peak points”including 1913 and 1993.

166

3. Mexican Oil Production. The scatterplot below shows total crude oil production forMexico from 1973 to 2003. Crude Oil Production - Mexico

050010001500200025003000350040001970 1975 1980 1985 1990 1995 2000 2005Thousands of Barrels Per Day

(a) If you looked only at the scatterplot, and did not consider any other information, whichfamily of functions, linear, exponential, or logarithmic, would you select as the mostappropriate to model this data? Explain why you picked as you did.

(b) The data below corresponds to the scatterplot above.

Year Production Year Production Year Production1973 465 1984 2780 1994 26851974 571 1985 2745 1995 26181975 705 1986 2435 1996 28551976 831 1987 2548 1997 30231977 981 1988 2512 1998 30701978 1209 1989 2520 1999 29061979 1461 1990 2553 2000 30121980 1936 1991 2680 2001 31571981 2313 1992 2669 2002 31771982 2748 1993 2673 2003 33711983 2689

i. Let t = 0 represent 1970. Find the linear regression model for this data.ii. Again, let t = 0 represent 1970. Find the logarithmic regression model for this data.

(c) Graph the two models from part (b) together with the data. Which model seems to givethe best fit.

(d) Why would a logarithmic model not be appropriate for modeling crude oil productionin the long run?

4. Benford’s Law. In 1938 Dr. Frank Benford, a physicist at the General Electric Company,noticed that, given a list of numbers, more numbers started with the digit 1 than any otherdigit. Random numbers are equally likely to start with a 1, 2, 3, ..., 9 yet that did notappear to be what really happened in lists of data. Dr. Benford analyzed 20,222 sets ofnumbers including areas of rivers, baseball statistics, numbers in magazine articles, and street


addresses. He concluded that the probability of the numeral d appearing as the first digitwas

log(

1 +1d

)

In other words, according to Benford’s Law, the probability that the first digit is a 1 islog(1 + 1) ≈ 0.30, the probability that the first digit is a 2 is log(1 + 1

2) ≈ 0.18, etc. So aboutthree-tenths of the numbers in a list typically start with the digit 1 (even though a randomlist of numbers should have only one-ninth of the numbers starting with the digit 1.)105

(a) Complete the following table.

d 1 2 3 4 5 6 7 8 9log(1 + 1/d)

This table, according to Benford’s law, gives the proportion of first digits that are ones,twos, etc.

(b) We looked at 60 retired baseball jersey numbers from the American League.106 Thefollowing table indicates how many numbers started with a 1, 2, 3, ..., 9.

First Digit ofJersey Number 1 2 3 4 5 6 7 8 9Frequency 17 11 12 5 5 2 2 3 3

Do these numbers seem to conform to Benford’s Law? Justify your answer by comparinga relative frequency distribution of the first digit of the jersey number to the distributionfrom part (a).

(c) We also looked at the populations in 1990 of counties in Michigan.107 The followingtable indicates how many numbers started with a 1, 2, 3, ...., 9. There are 83 numbersin all.

First Digit ofPopulation Number 1 2 3 4 5 6 7 8 9Frequency 26 17 10 4 9 2 7 5 3

Do these numbers seem to conform to Benford’s Law? Justify your answer by compar-ing a relative frequency distribution of the first digit of the population number to thedistribution from part (a).

(d) Using one page from a newspaper, list all of the numbers that appear. Do they seem toconform to Benford’s Law? Justify your answer.

(e) Benford’s Law is currently used in software designed to detect fraud and tax evaders.Explain how you think it does this and why this is likely to work.

5. Child Growth. The table below gives average weights for a sample of children in the U.S.

(a) Make a scatterplot of this data. Which family of functions, linear, exponential, orlogarithmic seems most appropriate to model this data and why?

(b) Find a logarithmic regression model for this data, and graph it along with the scatterplot.How well does the model seem to fit the data?

(c) Fill in the third column of the table, using your equation to predict the weights for eachof the given inputs.

105Browne, Malcom W. “Looking Out for No. 1.” New York Times 4 August 1998, p F4.106Baseball Almanac- The ”Official” Baseball History Site. http://www.baseball-almanac.com. 22 August 2002.107Time Series of Michigan Population Estimates by County. http://eire.census.gov/popest/data/counties/tables/CO-

EST2001-07/CO-EST2001-07-26.php. 23 August 2002.

168

Age in Weight Logmonths in pounds Model

3 12.36 16.29 19.112 21.115 22.818 2421 2524 25.8

(d) Use your model to predict weights for children who are 5 years old, 10 years old, and 15years old. Do these predictions seem reasonable?

(e) Use your model to predict the weight of someone who is 50 years old. Does this predictionseem reasonable?

(f) Why would this model not be appropriate to model the weights of infants less than amonth old?

6. Comparing Car Models III. This activity uses the same data on car models that appearedin chapters five and six. You can refer there for a table giving the complete data set.

(a) Make a scatterplot of greenhouse gas emissions versus city mileage.i. The exponential regression model for this data is y = 17.3(.963x) where y represents

the tons of carbon dioxide emitted per year by a car which gets x miles per gallonin city driving. Graph this equation along with your scatterplot. How well does itseem to fit?

ii. What does your exponential model predict for a car which gets 36 miles per gallonin the city?

iii. Using your exponential model, estimate the city miles per gallon for a car whichemits 4 tons of carbon dioxide per year.

iv. Repeat the previous part for a car that emits 3 tons of carbon dioxide per year.(b) Make a scatterplot of greenhouse gas emissions versus engine size. Note that engine size

is measured in liters.i. Explain why an exponential model would probably not provide a good fit to this

data.ii. Recall that the concavity of a graph indicates how the rate of increase or decrease in

the dependent variable is changing with respect to the independent variable. Thisscatterplot is slightly concave down, and increasing. What does the concavity sayabout the rate of increase?

iii. Find the logarithmic regression model for this data and graph it along with thescatterplot.

iv. What does your logarithmic model predict for a car that has an engine size of 1.3liters? What about 5 liters?

v. If you want to buy a car that only emits 4 tons of carbon dioxide per year, aboutwhat size of engine should you be considering?

(c) It is estimated that there are about 600 million passenger cars in the world.i. Assuming that these cars emit an average that is the same as the 38 cars in our sam-

ple, how many tons of carbon dioxide are being emitted each year by the estimated600 million cars?

ii. How much could this total be reduced if the average car emitted the same amountof carbon dioxide as a Toyota Celica?

8. Power and Quadratic Functions

It is the mark of an educated mind to be able to entertain a thought without accepting it.— Aristotle

In the last two chapters, we considered the properties of exponential and logarithmic functions,and mentioned power functions briefly. Recall that:

• Power functions are those which can be written in the form y = axb. The variable is a baseand the exponent is a constant.

• Exponential functions are those which can be written in the form y = abx. The variableis an exponent and the base is a constant.

In this chapter, we will examine power functions in greater detail, as well as a quadraticfunctions, which is a family related to the power function family with exponent b = 2.

Power Functions

As in the previous chapter, when considering examples of functions from a particular family,it is a good idea to experiment with your own particular examples so that you get a feel for howthat family of functions work. We have divided the examples of power functions into cases (or wemight say subfamilies), so that you need not consider all possible power functions all at once.

Example 1. Graph several examples of power functions y = axb, where b is a positive integer.Use a variety of values for the parameter a, although it is probably a good idea to try severaldifferent values of b for the same a as you begin. You might start with a = 1, varying b, andthen try a different a with the same set of b values. In each case, determine any x or y intercepts,decide whether the function is increasing or decreasing, and describe the concavity. Then, writea few sentences summarizing your results. Try to characterize the graphs of power functions withpositive integer exponents as completely as possible. How does the exponent effect the graphicalbehavior?

Solution: Figures 1 shows two examples for graphs of y = axb where b is a positive integer. Forx > 0, both graphs are concave up and increasing. What if x < 0? If b is an even integer, thenthe graph is decreasing and concave up. If b is an odd integer, the graph is increasing and concavedown. Both graphs go through the origin, and this is the only point at which the graphs crosseither of the axes. Also, we might notice that all the graphs go through the point (1, a), whateverthe parameter a happens to be (this fact is easy to miss, unless you try several examples with thesame a). You can see why this happens by substituting x = 1 into the general equation y = axb.

Finally, you might notice that the larger the value of b, the steeper the graph is for larger x. Infact, since all the graphs of y = axb go through (1, a), we can see that larger b give a steeper graphfrom this point on.

One final note. If you graph y = xb using a small window from x = 0 to x = 1, you will see thatthe larger the b, the closer the graph is to the x-axis. Why is this? Think about what happens ifyou raise a number between 0 and 1 to larger and larger powers (you could even try some exampleson your calculator). You could try x = 0.5, for example, and then raise this number to the secondpower, the third power, etc. The larger the exponent b, the smaller the resulting value of xb.

In the next example, we will consider negative values for b in y = axb. If you are usinga graphing calculator, you should change your graphing mode from Connected to Dot whengraphing functions with negative b. To see the reason for this, consider f(x) = 1/x. This functionis not defined for x = 0 (in other words, 0 is not in the domain of the function). We can evaluatef for x-values which are close to 0 (like .1,−.1, .01, etc.), and Table 1 shows the results from someof these evaluations.

The closer x is to zero, the larger (in absolute value) the value of 1/x. The graph of y = 1/x isshown below (Figure 2), and we see that the y-axis is a vertical asymptote for the graph. The

169

170

y = x 2

-2

0

2

4

6

-3 -2 -1 0 1 2 3x


y = x 3

-4

-2

0

2

4

-2 -1 0 1 2x


Figure 1: Two basic power function shapes.

x 5 0.1 0.01 0.0011/x 2 10 100 1,000x −0.5 −0.1 −0.01 −0.001

1/x −2 −10 −100 −1, 000

Table 1:

graph gets closer and closer to this line as x gets closer and closer to 0, but the graph never touchesthe axis.

When a graphing calculator (or other technology that is not designed to handle vertical asymp-totes) tries to graph a function like y = 1/x, it does not know there is a vertical asymptote, so itsimply plots a number of points and then connects the dots. If you set your calculator to graphin Dot mode, you will see just the points that the calculator is graphing, without the connectinglines, and this will give you a more accurate picture of the graph.

Example 2. Now, experiment with power functions y = axb with b a negative integer. Tryseveral examples, until you feel comfortable that you understand the general characteristics of thesefunctions, and then write a few sentences describing what you know.

Solution:Figure 2 shows two examples you may have considered, y = x−1 = 1/x and y = x−2 = 1/x2.

Example 3. Finally, let’s consider what happens if b is not an integer. Graph several examplesof y = axb, and use a variety of exponents b, some between 0 and 1 and some larger than 1, as wellas a variety of values for the parameter a. As before, it is a good idea to only vary one of theseat at a time (for example, you might graph several functions with different values of b but all witha = 1). In each case, determine any x or y intercepts, decide whether the function is increasing ordecreasing, and describe the concavity. Then, write a few sentences summarizing your results. Tryto characterize the graphs of power functions as completely as possible. How does the exponenteffect the graphical behavior?

Solution: Several examples are shown in Figure 3. Notice that, if the exponent b is not aninteger, the graph often exists only where x ≥ 0. We will see why this happens later.

For any b > 1, the graphs are concave up, and increasing for x > 0. If 0 < b < 1, then the graphis increasing and concave down. In both cases, the larger the value of b, the steeper the graph isfor x > 1. As in the previous cases of power functions, the graph of y = axb goes though (1, a).

8. Power and Quadratic Functions 171

–4

–3

–2

–1

0

1

2

3

4

y

–3 –2 –1 1 2 3

x

–2

–1

1

2

3

4

y

–3 –2 –1 1 2 3

x

(a) (b)

Figure 2: Two basic power function shapes for negative integer powers. Figure 2(a) is f(x) = 1x

and Figure 2(b) is g(x) = 1x2 .

f(x)

g(x)

h(x) i(x)

Figure 3: Graphs of f(x) = 2x0.7, g(x) = 4x0.5, h(x) = 2x2.5 and i(x) = 4x1.5

Domains of Power Functions

Why does the graph of y = axb exist only for non-negative x for many values of b whichare not integers?

Let’s consider f(x) = x0.5 as an example. Recall that

f(x) = x.5 = x1/2 =√

x

Since√

x is not defined for negative x, the domain of this function is x ≥ 0.108 So, the graphof f exists only for x ≥ 0. In general, if b is not an integer, the evaluation of f(x) = axb involvessome type of root, and many of these roots will not be defined for negative values of x. Thus, somegraphing technologies will not graph y = axb for negative x if b is not an integer.

108We can define√

x for negative x if we use imaginary or complex numbers, but in this course, we will only usereal numbers for domains and ranges of functions, unless otherwise noted.

172

Quadratic Functions

A quadratic function is a function of the form

f(x) = ax2 + bx + c.

Here, a, b, and c are parameters. We have already met one member of the family of quadraticfunctions, namely y = x2, in Figure 1. Another example is shown below in Figure 4. The shapeof the graph of a quadratic function is called a parabola. All quadratic functions have thisshape, although if the parameter a is negative, the shape will be turned “upside-down.” Thelow point (or high point) of a parabola is called the vertex. We will soon see how the threeparameters of a quadratic function determine the vertex, as well as the steepness or flatness ofthe corresponding parabola. Quadratic functions are used extensively in physics, and also haveapplications in business, economics, and other fields.

y = 2 x 2− 4x 5+ +

-2

0

2

4

6

8

x-2 -1 0 1 2 3 4


Figure 4: The parabola y = −2x2 + 4x + 5. Here a = −2, b = 4, and c = 5.

The Quadratic Formula

At some point in your mathematical studies, you have probably seen the so-called quadraticformula. This formula can be used to solve any equation of the form ax2 + bx + c = 0. Theformula says that solutions for x are given in terms of the parameters a, b, and c by

x =−b±√b2 − 4ac

2a

For a quadratic function f(x) = ax2 + bx + c, the solutions given by the quadratic formula tellyou the x-intercepts of the parabola, as long as the quantity under the square root sign is notnegative.109

One convenient property of parabolas is that they are symmetric about the vertical line throughthe vertex. So, whenever there are two intercepts, the x-coordinate of the vertex will be half way in

109If the quantity b2− 4ac under the root sign is negative, then the solutions to the equation are complex numbers,and will not correspond to any points on the graph of the parabola. The expression b2−4ac is called the discriminantof the equation.


between the x-intercepts. Because of this, one can deduce that the vertex of a quadratic functionwill be the point

(−b

2a, f

(−b

2a

))

To see why this is, note that the quadratic formula can be written

x =−b±√b2 − 4ac

2a=−b

2a±√

b2 − 4ac

2a

This means that one x-intercept is equal to −b/2a plus a particular number, given by the expressionafter the ± in the equation above. The other x-intercept is −b/2a minus this same number. So,the two x-intercepts are the same distance away from −b/2a which means −b/2a must be half wayin-between the two intercepts.

You can use the quadratic formula and the vertex formula to easily relate the parameters ofany quadratic function to its x-intercepts and vertex.

Example 4. Find the vertex and x-intercepts of the graph of y = 2x2 − 8x + 3.

Solution: The parameters are a = 2, b = −8, and c = 3. So, −b2a = 8

4 = 2. We havef

(−b2a

)= f(2) = 2 · 22 − 8 · 2 + 3 = −5. So, the vertex is (2,−5).

The x-intercepts can be found using the quadratic formula and are

x =−b

2a±√

b2 − 4ac

2a= 2±

√(−8)2 − 4 · 2 · 3

4= 2±

√404

≈ 2± 1.58

or about 0.42 and 3.58.

From this chapter and the last, we now have a fairly good idea of the general characteristicsof graphs for exponential, logarithmic, power, and quadratic functions, based on observations ofgraphs. We also have some connections between the symbolic and the graphical representations.For exponential functions y = abx, the base tells us whether the graph is increasing or decreasingand how steeply it does so. The parameter a tells us the y-intercept. For power functions y = axb,the exponent b determines where the function is concave up and concave down as well as thesteepness.

Knowing these general characteristics will help us decide which family is more likely to give usa good mathematical model when we are presented with a scatterplot of data.

Power Regression Models

In Chapters Six and Seven, we used exponential and logarithmic regression to find models.Similarly, we can use power or quadratic regression to find models when either of these families offunctions seems appropriate. This can be done on a TI calculator or in Excel.

It is important to remember when using the power regression feature in any technologyto exclude any zero or negative values in either the independent or dependent variablelists! The reason for this is that the domain of log x is all positive real numbers. Wewill discuss the details of why this is in the next section of this chapter.

Example 5. In Chapter Two, we introduced Lorenz curves and the Gini coefficient. In theActivities, you may have looked at the Lorenz curves for several countries, including the U.S.Figure 5 shows the Lorenz curve, and the data on which it is based is in Table 2. The data is forthe U.S. for the year 1989. Recall that the Gini coefficient is twice the area between the curve andthe “equality line.”

174

Table 2: Distribution of Income by Percentile for the U.S., 1989

Population Percentile 0 0.20 0.40 0.60 0.80 1.00Percent of Income 0.000 0.046 0.152 0.317 0.554 1.000 Lorenz Curve for U.S. - 1989

0.000.100.200.300.400.500.600.700.800.901.000 0.2 0.4 0.6 0.8 1Cumulative Percent of PopulationCumulative Percentage

of Income USAEqualityFigure 5: Lorenz Curve for U.S., 1989

In the Chapter Seven Activities, you may have found power functions of the form L(x) = xb tomodel this data by finding b so that the graph of y = L(x) went through one of the points for whichx was not 0 or 1, in addition to the two points (0, 0) and (1, 1). The problem with these functionswas that, even though they went exactly through three of the points, they were somewhat far awayfrom some of the other points. Can we find a power function which is “on average”, closer to allthe points?

Solution: To do this, we will find the power regression function modeling this data andusing technology, for example a TI-83 calculator.110 We simply input the x-values .2, .4, .6, .8, and1.0 into one list, and the y-values (as shown in the table above) in another list (but not y = 0).Remember that we cannot use 0 as an x-value or a y-value when doing power regression. The powerregression feature can be found in the STAT menu, under CALC. Then scroll down to PwrReg.The calculator returns the two parameters which define the power function. The resulting functionis y = 0.89x1.88. Figure 6 shows the scatterplot for the data in Table 2, values for a Lorenz curve,along with our regression function and the function L(x) = x2.25. The regression equation seemsvisually to be a better fit for the data, except near the point (1, 1).

Re-expression and Nonlinear Regressions

Let’s look at exactly how your TI calculator or whatever technology you have available iscalculating the exponential and power regression functions. As we do this, we will explain someof the problems with non-linear regressions. Understanding the mathematical details of how non-linear regression works will give you a better understanding of how to use these models, and also

110Excel does not have a built in power regression feature. However, we will eventually see how we can use Excel’slinear regression feature to find the power regression function.

8. Power and Quadratic Functions 175 Lorenz Data and Two Approximations for U.S. - 19890.000.100.200.300.400.500.600.700.800.901.00

0 0.2 0.4 0.6 0.8 1Cumulative Percent of PopulationCumulative Percentage of Income USAL(x)Power Reg

Figure 6: Scatter plot of Lorenz information with two power models

help you make judgements on which non-linear model is the most appropriate. In addition, it willgive us an example of the use and importance of logarithms and their properties.

In Chapter Five, when we introduced linear regression, we noted that the linear regression lineis the line which minimizes the sum of the squared errors (refer back to that chapter to review thedetails).111 The calculation depends on the fact that we are using a linear function, the simplesttype of function, to model the data. In cases where the data is not linear, and we want to find anon-linear model, we will first need to re-express the data to see if we can “make it approximatelylinear”, and then use linear regression.

We will first explain the steps a calculator or other piece of technology would do internally tofind the exponential regression, and then explain why this process works to give a good exponentialmodel. We will then do the same for power regression.

Exponential Regression and Re-expression

Figure 7 shows the Child Mortality Rate for Albania over time. Obviously the Child MortalityRate will always be at least 0. The scatterplot is decreasing, and the trend seems to be leveling offtowards zero over time. The general shape is concave up. All these indicate that an exponentialmodel with base between 0 and 1 might be appropriate.

To find the exponential regression function for this data, your technology would go through thefollowing steps (but of course, automatically and very quickly).

1. The technology takes the logarithm of the given dependent variable values, in this case the(Child) Mortality values. The Mortality Rates along with Log(Mortality) are shown in Table3.

2. The technology finds the linear regression line for Log(Mortality) versus Year data. Figure 8shows a scatterplot of Log(Mortality) versus Year along with the graph of the linear regressionline for this relationship. Notice that the line does seem to fit the data fairly well. Theequation for the regression line y = Mx+B shown has slope M = −0.016455 and y-interceptB = 2.12646. We are using capitals M and B to denote the parameters for this linearequation to avoid confusion with the parameters we will get for the exponential model below.

111The details of how the linear regression calculation is done can be found in most elementary statistics books.

176 Child Mortality Albania (Deaths per 1000 live births)020406080

1001201401601950 1960 1970 1980 1990 2000 2010

Figure 7: Child Mortality Rates for Albania over time

Year Childfrom 1960 Mortality Log(Mortality)

0 151 2.17897710 82 1.91381420 57 1.75587530 45 1.65321335 35 1.54406840 31 1.491362

Table 3: Mortality and Log(Mortality) as a function of Year since 1960

3. The technology finds the parameters a and b for an exponential equation y = abx modeling theoriginal Mortality versus Year data. It does this by converting the linear regression equationfor Log(Mortality) versus Year into an exponential model for the the original Mortality versusYear relationship. This is done using basic properties of exponents and logarithms.

• The parameter a, which is the y-intercept for the exponential model, equals 10B, whereB is the y-intercept for the linear model for the logarithm data.

• The parameter b, which is the base for the exponential model, is equal to 10M , whereM is the slope for the linear model of the re-expressed (logarithmic) data.

In our case, we have a = 10B ≈ 102.12646 ≈ 133.8 and b = 10M ≈ 10−0.016455 ≈ .9628. Theexponential equation y = 133.8(.9628x) is shown below along with the original scatterplot inFigure 9. The correlation coefficient for the exponential regression given by the technologyis r = 0.988.

Why does this process work?

When we find the linear regression model for Log(Mortality) versus Year, we are really getting amodel of the form log y = Mx+B, where y will give the predicted values for the original Mortalitydata. Using basic algebra, including the properties of logarithms we considered in the previouschapter, we can transform this equation into an exponential model for y versus x. Recall that

8. Power and Quadratic Functions 177Log of Child Mortality Versus Year - Albania00.511.522.51950 1960 1970 1980 1990 2000 2010

Figure 8: A plot of the logarithms of child mortality versus year, with linear regression lineChild Mortality in Albania with Exponential Regression Curve0204060801001201401600 10 20 30 40 50

Figure 9: Child Mortality in Albania with Exponential Regression Model

log y = x means the same as 10x = y

So, applying this to the equation log y = Mx + B, we get

log y = Mx + B means the same as 10Mx+B = y

Using properties of exponents we get

y = 10Mx+B = (10Mx)(10B) = (10M )x(10B) = 10B(10M )x

If we let a = 10B and b = 10M as above, this becomes y = abx.

Power Regression and Re-expression

For power regression, your technology also uses re-expression. In this case, if we let Independentand Dependent represent your data variables, then your technology lets X = Log(Independent)

178

and Y = Log(Dependent) and then finds the linear regression model for X and Y . This results ina model of the form.

log y = M log x + B

Where x and y are the variables modeling the original Dependent versus Independent datarelationship. Changing to exponential form as before, we get

y = 10M log x+B = (10M log x)(10B)

Now, from our properties of logarithms, M log x = log xM . Also, recall from our properties oflogarithms that 10log xM

= xM . From these, we get that

y = (10M log x)(10B) = (10log xM)(10B) = xM · 10B = (10B)xM

so, the parameters a and b for the power regression for y versus x are given by a = 10B and b = M .Also, we can now say why you cannot use 0 in your data set for either the dependent or

independent variable when doing power regression. In fact, you also cannot use negative numbers.Recall the log x is not defined unless x is positive. If your data has 0 or negative values for theindependent variable (or x), then your technology will give an error when it tries to take thelogarithm of these values. The same thing happens if you have any dependent variable values thatare zero or negative.

With exponential regression, the technology is not taking the logarithm of the independentvariable values, but it is taking the logarithm of the dependent variable values so you will typicallynot be able to do exponential regression on data where there are any dependent variable (or y)values which are negative or 0.

Correlation Coefficients for Non-linear Regressions

Above, we cautioned that the linear regression correlation coefficient cannot be compared to theexponential (or other non-linear) correlation coefficient. We will illustrate why using the AlbanianChild Mortality Data above.

Recall that the linear regression line is the line that minimizes the sum of the squared errors,where the errors are the differences between the actual data values for the dependent variable foreach x, and the y-values predicted by the linear regression line for each x. We might hope thatthe exponential regression equation would give the smallest sum of squared errors over all possibleexponential models. However, this is not the case. Table 4 gives the original data along with thevalues predicted by the exponential regression equation y = 133.8(.96282x), the values predictedby another exponential model f(x) = 151(.9591x), as well as the errors between these modelsand the data, the squared errors, and the sum of the squared errors. Note that the sum of thesquared errors for the exponential regression equation is 427.308, which is larger than the sum ofthe squared errors for f(x), 387.018. If we measure “best” by which function gives us the smallestsum of squared errors, the exponential regression function is not the best exponential model.

If the exponential regression model is not the best in this sense, how is it the best and why dowe use it?

To answer the second question first, the technology uses the regression procedure because itreduces to the known linear regression procedure. Finding the nonlinear function that minimizesthe sum of squared errors is possible, but more involved. The non-linear regressions are the bestlinear fit to the corresponding re-expressed data. Thus, they provide a reasonably good model inmost cases, even if it is not necessarily the best we can do.

Logarithmic Regression

Some technologies will also do logarithmic regression. This type of regression gives you the“best” logarithmic model for a given two-variable data set. These models would be of the form


Table 4: Comparison of Exponential Regression Errors with a Second Exponential Model

Child Exponential Regression Errors Squrd. Reg. Squrd. Err.Year Mort. Regression 151(.9591x) Errors 151(.9591x) Errors 151(.9591x)

0 151 133.800 151.000 -17.200 0.000 295.840 0.00010 82 91.583 99.453 9.583 17.453 91.842 304.59120 57 62.687 65.502 5.687 8.502 32.342 72.28530 45 42.908 43.141 -2.092 -1.859 4.376 3.45535 35 35.499 35.012 0.499 0.012 0.249 0.00040 31 29.370 28.414 -1.630 -2.586 2.658 6.687

Sum of Err2 427.308 387.018

y = a log x + b or y = a ln x + b

depending on whether the technology is using the common base 10 logarithm or the base e naturallogarithm. As with exponential and power regressions, re-expression is involved. In this case, there-expression takes the logarithm of the x or independent variable values.

Applications of Power Models

Example 6. Table 5 shows how life expectancies changed in the United States during the 20thcentury. These same data were also considered in Chapter Seven, where we found a logarithmicmodel.112 The value given for each year represents how long a person born in that year couldexpect to live, on the average. Find an appropriate power model for this data, and discuss howgood the fit is. Use the model to predict the life expectancy for U.S. children born in 2010 and2050. How reasonable do these predictions seem to be?

Year 1900 1915 1930 1945 1960 1975 1990 2000Life Expectancy 47.3 54.5 59.7 65.9 69.7 72.6 75.4 76.4

Table 5: Life Expectancy for U.S. children born from 1900 to 2000.

Solution: Figure 10 shows a scatterplot of the data. The pattern seems to be slightly con-cave down. This indicates that a power function with exponent between 0 and 1, as well as thelogarithmic model we had previously found, might be appropriate.

To test if either is appropriate, and which one might be better, we do what is called “re-expressing the data.” If the plot of Log(Life Expectancy) versus Log(Year) is approximately linear,then a power function would be a good model. If the plot of Life Expectancy versus Log(Year)is fairly linear, then a logarithmic function would make a good model. If we let the independentvariable be years from 1900, however, we will have to take into account that we cannot use 0 as avalue when taking logarithms. To adjust for this, we leave off the first data point and constructthe plots using only the years starting with 1915, which corresponds to an x-value of 15. Bothplots are shown in Figure 11, along with the linear regression line for these relationships.

Both plots seem to give a very strong linear pattern, which means either type of function shouldmake a good model. The Log(y) versus Log(x) seems to give a slightly stronger linear relationship,so the power model might be the best. We can do both a power regression and a logarithmicregression, either using technology that does this automatically, or by re-expressing the data in the

112This example is adapted from Functioning in the Real World by Sheldon Gordon, et. al., p. 189

180 U.S. Life Expectancy from 1900404550556065707580

0 15 30 45 60 75 90 105Figure 10: U.S. Life ExpectancyLog Y vs Log X plot to test for Power Relationship

1.71.751.81.851.91 1.2 1.4 1.6 1.8 2 2.2

Y vs Log X plot to test for Logarithmic Relationship50556065707580

1 1.2 1.4 1.6 1.8 2Figure 11: Plots of re-expressed life expectancy data

table and using linear regression on the re-expressed data (again, leaving out the year 1900). Weget the following models:

• The power model is y = 32.66(x0.1842)

• The logarithmic model is y = 12.05 ln x + 20.521

The original data with each of these models is graphed in Figure 12. Both seem to fit the datawell.

For the year 2010, the power model predicts a life expectancy of 77.64, while the logarithmicmodel predicts 77.02. These are quite close to each other. For the year 2050, the power projection is82.20, while the logarithmic projection is 80.73. The power function is providing a higher estimateas time goes on. This should not be surprising, as power functions, even when concave down, growmore quickly than logarithmic functions.

8. Power and Quadratic Functions 181U.S. Life Expectancy Data with Power Model404550556065707580

0 15 30 45 60 75 90 105U.S. Life Expectancy Data with Log Model

4045505560657075800 15 30 45 60 75 90 105

Figure 12:

Modeling Revisited

Figure 13: The Mathematical Modeling Process

In Chapter Four, Functions, we defined a modeling function as one that describes a real worldphenomenon. However, a modeling function often only gives an approximation of the real worldsituation. In general, a mathematical model is a representation of a particular phenomenon whichattempts to take into account the most important characteristics of that phenomenon, while ignor-ing irrelevant details. This means that models are not often (perhaps almost never) a completelyaccurate description of the phenomenon. They are simplified versions of reality.

In fact, in Chapter Five, Linear Functions, we commented that one reason linear functionsare so important is that they are the simplest types of functions. The two quotes that beganthat chapter highlight the notion that one should always pick the simplest model that adequatelyexplains the given situation. Oftentimes, one can create a better model than the current model,but this often means making the model more complicated. Inherent in all modeling is this trade-offbetween quick and easy, but perhaps over-simplified models, and complicated, hard, but perhapsmore accurate and complete models.

In practice, models are often created using an iterative process like the one in Figure 13.

182

Two Kinds of Models

When we considered linear models, there were actually two different approaches we took tocreating models. In Example 3 of Chapter Five, we considered a furnace repair person who charges$20 to visit your house, and then $30 per hour for his work. The function which models the totalcharge for a visit is C(n) = 30n+20, where n represents the number of hours and C represents thecost. This type of model is called a theoretical model. It is based on known facts or assumptionsabout the relationship between the hours worked and the cost. We end up with a linear modelbecause we are told the rate of change measured as cost per hour is constant.

The second approach we took involved creating linear models using linear regression. Forexample, in Table 5 and Figure 6 in Chapter Five, we looked at data relating the Arm Span andHeight of a number of college students. In this case, we started with a set of data, and did notknow ahead of time whether or not the relationship between the two variables involved a constantrate of change. We simply took the data we had and found the linear equation that “best fit” thedata, where best is defined by minimizing the sum of the squared errors. Regression models areone kind of what are called empirical model. In this chapter and the last, we used regressionto find “best fit” non-linear models from a given function family for a given set of data. This issometimes called curve-fitting.

As we proceed, we will have even more options for creating models for a given set of data. Thechoice of which family (or combination of families) to use is often more of an art than a science.We will continue to try and create “the best” models we can, but will always need to acknowledgethe inherent limitations of any model.

Reading Questions for Power and Quadratic Functions

The function of education is to teach one to think intensively and to think critically ...Intelligence plus character - that is the goal of true education.— Martin Luther King, Jr.

1. Below are the graphs of y = x−2, y = x−1/2, y = x1/6, y = x1/2, y = x2 and y = x6. Identifyeach, briefly justifying your answer.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 1 2 3

0

0.5

1

1.5

2

2.5

3

3.5

0 2 4 6 8 10

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2

(a) (b) (c)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 2 4 6 8 10

0

0.5

1

1.5

2

2.5

3

3.5

0 1 2 3 4

0

2

4

6

8

10

12

14

16

18

0 1 2 3 4

(d) (e) (f)

2. If you are given a power function whose graph is concave up in the first quadrant, what doyou know about the exponent?

3. If you are given a power function whose graph is concave down in the first quadrant, whatdo you know about the exponent?

4. Let f be a power function of the form f(x) = axb.

(a) Suppose f(1) = 5. What is the value of a?(b) Suppose f(1) = 2 and f(2) = 8. What is the value of b?(c) Suppose f(1) = 5 and f(2) = 190.27 (rounded to two decimal places). What is the value

of b?

5. Let f be a function whose y-intercept is 5. How do you know that this cannot be a powerfunction?

6. Solve the following equations. Some of these are exponential equations, and some are powerequations.

(a) 23 = 3.6(1.2x)(b) 23 = 3.6(x1.2)(c) 22, 500 = 3, 000(1.005)12x

(d) 17 = 2.5(1 + x)3

183

184

Figure 14:

7. For each of the functions given in graphical form in Figure 14, decide whether the functionis linear, exponential, power, logarithmic, or none of these.

8. Solve the equations.

(a) x2 − 4x + 3 = 0

(b) x2 − 8x + 4 = 0

9. For each verbal description, decide which type of function, linear, logarithmic, exponential,or power, is most likely to fit the given information.

(a) The U.S. population has been growing at a rate of about .7% per year.

(b) World energy consumption has been growing at a constant rate of about 6 quadrillionBTU’s per year since 1980.


(c) f is concave down and increasing, with a y-intercept of about 0.(d) f is concave down and increasing, with an x-intercept of about 1.(e) f is concave up and decreasing, with a y-intercept of about 5.(f) f is concave up and decreasing, with no y-intercept.

10. In Chapter Eleven, one of the activities includes creating a model for the total amount ofenergy consumed worldwide starting in the year 1980. One possible model, letting n = 0represent the year 1980, is

C(n) =n(6n + 560)

2= 3n2 + 280n

where C(n) represents an estimate of total world energy consumption from 1980 through theyear represented by n in quadrillions of BTU’s.

(a) Estimate the year in which the total consumption from 1980 through that year will be10,000 quadrillion BTU’s.

(b) Estimate the year in which the total consumption from 1980 through that year will be40,000 quadrillion BTU’s. (40,000 quadrillion BTU’s was the U.S. Energy Department’sestimate of total known energy reserves, including coal, oil, and gas, as of 2002.)

11. The table below shows total U.S. energy consumption, in quadrillions of BTU, for the years1949 through 2002. Years are counted from 1900.

Yr Cons. Yr Cons. Yr Cons. Yr Cons.49 32.0 63 49.7 77 78.0 91 84.550 34.6 64 51.8 78 80.0 92 85.951 37.0 65 54.0 79 80.9 93 87.652 36.8 66 57.0 80 78.3 94 89.353 37.7 67 58.9 81 76.3 95 91.254 36.6 68 62.4 82 73.2 96 94.255 40.2 69 65.6 83 73.1 97 94.756 41.8 70 67.8 84 76.7 98 95.257 41.8 71 69.3 85 76.4 99 96.858 41.7 72 72.7 86 76.7 100 98.959 43.5 73 75.7 87 79.2 101 96.360 45.1 74 74.0 88 82.8 102 97.461 45.7 75 72.0 89 84.962 47.8 76 76.0 90 84.6

(a) Use logarithmic regression to find a logarithmic model for this data. Logarithmic re-gression gives you a model of the form y = m log x + b, with parameters m and b.

(b) Graph the data along with your logarithmic model. How well does the model seem tofit the data?

(c) Is there a different family that you think would provide a better model? Why do youthink so?

(d) Using the family you selected in part (c), find a model from this family that you thinkbest fits the data. How does the fit of this model seem to compare to the fit of thelogarithmic model?

186

12. Solve the following equations, where f(x) = 2.76x1.4

(a) f(x) = 0(b) f(x) = 1(c) f(x) = 2.76(d) f(x) = 10

13. Solve the following equations.

(a) 2x2 − 8x = 7(b) 2x3 − 4 = 9(c) 2(3x)− 4 = 9

14. Solve the following quadratic equations.

(a) x2 + 3x + 2 = 0(b) −16t2 + 85t + 5 = 0(c) 0.35x2 − 2.4x + 4.7 = 0(d) 3x2 − 2x = 7

15. The function h(t) = −16t2 + 85t + 5 represents the approximate height of a baseball thrownup in the air at 85 feet per second from a height of 5 feet.

(a) Evaluate h(0), h(1) and h(2).(b) Use the formula for the vertex of a parabola in the text to find the vertex of this parabola.(c) Interpret the vertex coordinates in terms of the flight of the ball. What does the x-

coordinate of the vertex tell you in this situation? What does the y-coordinate tellyou?

(d) According to this model, when will the ball hit the ground?(e) Suppose the ball was thrown from a height of 10 feet at 70 feet per second. How do you

think this would change the symbolic form for h(t)?

16. f(x) = ax2 + bx + c is the general form for a quadratic function.

(a) What does the parameter a tell you about the graph of the function? Experiment witha few examples as necessary.

(b) What does the parameter c tell you about the graph of the function?


Power and Quadratic Functions: Activities and Class Exer-cises

1. Baby Weights. The following table gives the average weight for babies according to theNational Center for Health Statistics.

Age (months) 3 6 9 12 15 18 21 24Weight (lb) 12.3 16.2 19.1 21.1 22.8 24.0 25.0 25.8

(a) Make a graph of weight versus age. (Your graph should have age on the horizontal axis.)(b) By looking at your graph, explain why a power function should do a better job of fitting

these data than an exponential function.(c) Using power regression, find a power function that fits the data where age is the input

and weight is the output.(d) Plot your power function along with the data. How well does your function fit the data?(e) The average weight of a one month old baby is 9.7 pounds. Using your power equation,

what is the average weight of a one month old baby?(f) The average weight of a newborn baby is 7.9 pounds. What does this imply about the

appropriateness of a power function for modeling baby weights over time (at least forvery young infants)?

(g) The following table gives the average weight of children ages 3 through 10.

Age (years) 3 4 5 6 7 8 9 10Weight (lb) 32 36 40 46 51 57 63 70

Plot these data along with your power function and the original data from part (d). Indoing so, remember to convert the years to months. For example, the first new pointshould be 36 months with a weight of 32 pounds.

(h) How well does your function fit the new data points from part (g)? How does thiscompare with the fit of the original data points?

2. Lorenz Curves and Power Functions. In Chapter Two, we introduced Lorenz curvesand the Gini coefficient. In one of the activities in that chapter, you may have looked at theLorenz curves for several countries, including the U.S. The graph below shows the Lorenzcurve, based on the data in the table which is for the U.S. for the year 1989.

Table 6: Distribution of Income by Percentile for the U.S., 1989

Population Percentile 0 0.20 0.40 0.60 0.80 1.00Percent of Income 0.000 0.046 0.152 0.317 0.554 1.000

The Lorenz curve is concave up, and goes through the points (0, 0) and (1, 1). This informa-tion we have is clearly consistent with a power function. In fact, since (1, 1) is on the graph,we might use a power function of the form L(x) = xb.

(a) As a first model, find a b which gives a power function L1(x) = xb going through thepoint (0.40, 0.152). If this point is on the curve of L1(x) = xb, then 0.152 = 0.4b. Uselogarithms to find b.

(b) Find power functions L2 and L3 which go through the points corresponding to x = 0.6and x = 0.8.

188 Lorenz Curve for U.S. - 19890.000.100.200.300.400.500.600.700.800.901.00

0 0.2 0.4 0.6 0.8 1Cumulative Percent of PopulationCumulative Percentage of Income USAEquality

Figure 15: Lorenz Curve for U.S., 1989

(c) Graph a scatterplot of the data in the table along with all three power functions L1, L2,and L3. Which function seems to be the best model?

3. Power Models for Lorenz Curves.

The table below shows Income Inequality Data for the Soviet Union, similar to the datashown in Example 5 for the United States where we were modeling the Lorenz curve for theU.S.

Table 7: Distribution of Income by Percentile for the Soviet Union, 1989

Income Percentile 0 0.2 0.4 0.6 0.8 1Percent of Population 0.00 0.0891 0.2260 0.4050 0.6340 1.0000

(a) Find the power regression model for this data, as in Example 1.(b) Graph your power regression model along with a scatterplot of the data. How well does

your model fit the data?(c) Graph a scatterplot of Log(Percent of Population) versus Log(Income Percentile). Does

this graph seem to have a strong linear relationship? What does this tell you about theappropriateness of a power model for this data?

(d) Find a second power model, one of the form y = xb. Models of this form will go throughboth the origin and the point (1, 1). Experiment with various values of b until you feelyou have the best fit you can. Do you think this model fits the data better than yourfirst model? Why or why not?


4. Child Mortality Models.The table and graph below show Child Mortality Rates for children under 5, in deaths per1000 live births, for four countries from 1960 to 2001.

Year Haiti Kenya Congo Laos1960 253 205 220 2351970 221 156 160 2181980 195 115 125 2001990 150 97 110 1631995 137 111 108 1342000 125 120 108 1052001 123 122 108 100Child Mortality Time Series

501001502002503001950 1960 1970 1980 1990 2000 2010

HaitiKenyaCongoLaosFigure 16: Childhood Mortality Time Series for Four Countries

(a) For the plots for Congo and Kenya, discuss why none of the four families linear, ex-ponential, power, or logarithmic would give an ideal model. In other words, for eachplot and for each of the four families, give at least one characteristic of the line plot(intercepts, asymptotes, etc.) that would not be a characteristic of the family.

(b) Which family of functions do you think would be most appropriate to model the graphfor Haiti and why?

(c) For each of the four countries, give your best estimate for what the child mortality ratewill be in the year 2020. One way to do this is to create models for each country, selectingan appropriate family given the scatterplot. Whatever method you use, carefully justifyyour estimate for each country.

5. Crude Oil Production Models. As noted in Chapter Two, fossil fuel supplies are finite.The U.S. has already passed its peak production years, and is now on the decline. Manyother countries are past their peak as well. With the quadratic family of functions, we finallyhave a family that can model phenomenon which have this kind of peak.The table below gives primary energy production from 1980 for three countries. This pro-duction includes crude oil, natural gas, and coal. The units are in quadrillions of BTU.

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Year from 1980 Egypt Norway Sweden0 1.453 3.012 0.8731 1.481 3.065 0.9882 1.691 3.062 0.9543 1.841 3.422 1.0714 2.080 3.638 1.2105 2.233 3.787 1.3096 2.093 3.947 1.3187 2.299 4.392 1.4048 2.238 4.764 1.3909 2.320 5.718 1.39110 2.361 5.942 1.41411 2.396 6.230 1.40812 2.444 7.079 1.39213 2.547 7.268 1.37314 2.594 7.643 1.32815 2.673 8.351 1.39016 2.729 9.280 1.26417 2.600 9.588 1.39718 2.569 9.347 1.49419 2.673 9.531 1.45820 2.628 10.212 1.38921 2.782 10.217 1.54322 2.719 10.610 1.36223 2.705 10.402 1.211

(a) Create scatterplots of production versus year for each of the three countries.i. For which two of the three countries does it appear a quadratic model would fit the

data well? Why?ii. For the third country, why does it appear a quadratic model would not fit well?

(b) For each of the two countries for which you believe a quadratic model would fit well,find the quadratic regression model for the production in this country.

i. Graph your models together with the corresponding scatterplots. Which modelsseem to fit well?

ii. For each country, use the model for that country to predict the energy productionin the years 2010, 2020, and 2050. Do these estimates seem to make sense? Whyor why not?

6. Kepler’s Laws: The table below shows the period in earth days, and the average distancefrom the sun in millions of miles for all nine “traditional” planets.

Planet Period DistanceMercury 88 36.0Venus 225 67.2Earth 365 92.9Mars 687 141.5

Jupiter 4,329 483.3Saturn 10,753 886.2Uranus 30,660 1782.3Neptune 60,150 2792.6Pluto 90,670 3668.2


(a) Notice the earth is 92.9 million miles from the sun and takes 365 days to orbit the sun.i. Find the distance the earth travels around the sun (in miles) and then use this to

find how fast the earth is traveling in miles per hour.ii. Find out how fast Jupiter is traveling in miles per hour as it orbits the sun.

(b) Graph a scatterplot of these data. Which family of functions do you think would bemost appropriate to model these data? Explain why you think so.

(c) Find the power regression model for the data. How well does the model fit the data?(d) Astronomers have recently been discovering large objects that may qualify as planets

past Pluto. One of these has been named Quaoar (pronounced kwah-o-wahr). Quaoar’speriod is estimated to be about 285 earth years. Use your power model to estimate howfar Quaoar is from the sun, on the average.

(e) In the 1950’s, Dutch astronomer Jan Oort conjectured that there was a massive sphereof comets orbiting the sun at distances around 4 trillion miles. The existence of thisso-called “Oort cloud” has since been confirmed. Using your power model, estimate theperiod of objects in the Oort cloud. Give your answer in Earth years.

(f) Using your answer from part (e), about how fast are objects in the Oort cloud travellingin miles per hour?

9. Transformations of Functions

Nothing has afforded me so convincing a proof of the unity of the Deity as these purelymental conceptions of numerical and mathematical science which have been by slowdegrees vouchsafed to man, and are still granted in these latter times by the DifferentialCalculus, now superseded by the Higher Algebra, all of which must have existed in thatsublimely omniscient Mind from eternity.— Mary Somerville

The graph in Figure 1 shows the relationship between survival of female children to age 10and the illiteracy rate for a number of countries in North and South America. Both variables aremeasured in percent. The trend is definitely decreasing, but it is not clear from the plot whethera linear or non-linear model would be more appropriate. There are three points which could beconsidered outliers.

If we were to model this data empirically, we might try both a linear model and a decreasingexponential model and use our own “visual judgement,” correlation, and perhaps an analysis ofthe sum of squared errors to decide which was a better fit. Leaving out the outliers would allowus to get a much better model for the remaining points.

On the other hand, we might instead think about making some assumptions about this rela-tionship, and use these to create a theoretical model, or at least a model that is partially basedon theoretical assumptions.113 Percent of Females Surviving to Age 10 Versus Illiteracy Rate - Americas

889092949698100

0 15 30 45 60Figure 1: Survival of Female Children to Age 10 Versus Illiteracy Rate

What are some reasonable assumptions might we make?

Well, we obviously can’t have a survival rate higher than 100% or lower than 0%, and thesame goes for the illiteracy rate. From the scatterplot, we might assume that a 0% illiteracy ratecorresponds to a 100% survival rate (or perhaps 99.9% or something close to 100%), and createour model accordingly. Also, although it is conceivable you might have countries with illiteracyrates ranging from near 0% to near 100%, it is very doubtful that the survival rate would ever gettoo close to 0%. We might assume that there is some lower bound on how low the survival ratecan be, and that it will never go lower than this bound. Of the countries in our scatterplot, nonehas a female survival rate to age 10 lower than 89%.

113We discussed empirical and theoretical models briefly at the beginning of the last chapter.

192

9. Transformations of Functions 193

Given these assumptions, we might want a model which is decreasing and concave up, andlevels off at whatever lower bound seems reasonable. Of the models we have looked at so far(linear, exponential, power, and logarithmic) none would satisfy all these criteria. An exponentialmight work, except that decreasing exponential functions level off at a lower bound of y = 0. Whatmight work is to have a model which has the same shape and characteristics of an exponentialfunction, but levels off somewhere besides 0. We can in fact create such a model using exponentialfunctions and what are called transformations of functions.

We will consider four basic types of transformations to start with. We will introduce theseusing graphical representations, after which we will show how to transform a function using itssymbolic representation. The four types of transformations are:

1. Vertical shifts: The graph is moved up or down, without distorting the shape.

2. Horizontal shifts: The graph is moved left or right, without distorting the shape.

3. Vertical stretches and compressions: The graph is stretched or compressed (squashed)in the vertical direction. When this is done, points on the x-axis do not move, and points offthe x-axis are moved away from (for a stretch) or towards (for a compression) the x-axis.

4. Horizontal stretches and compressions: The graph is stretched or compressed (squashed)in the horizontal direction. When this is done, points on the y-axis do not move, and pointsoff the y-axis are moved away from (for a stretch) or towards (for a compression) the y-axis.

The following “fun house mirror” pictures illustrate a couple examples of these transforma-tions.114

(a) (b) (c)

Figure 2: Image (a) represents an undistorted graph. Image (b) represents a horizontal compression.Image (c) represents a vertical compression.

Now, let’s relate the visual effects of these transformations to what is happening symbolically.

Transformations of Functions from the Symbolic Point of View

Vertical shifts: Adding a Constant to the Output

Consider f(x) = 1−x2 and let g(x) = f(x)+5. Note that we can write g(x) as g(x) = (1−x2)+5.For each input x, the output or y-value for g is 5 more than the output for x. Graphically, thismeans that the graph of g will be the same shape as the graph for f but 5 units higher over the

114Original photo of De Zwaan taken by the author in Holland, Michigan.

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whole domain. For instance, if x = 1, then f(1) = 1 − 12 = 0, so the point (1, 0) is on the graphof f . We have, g(1) = (1− 12) + 5 = 0 + 5 = 5, so the point (1, 5) is on the graph of g. The point(1, 5) lies 5 units directly above the point (1, 0). Figure 3 shows graphs for both y = f(x) andy = g(x) = f(x) + 5.

y = 1 x 2−

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y = 1 x 2−( ) 5+

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Figure 3: Graph of y = 1 − x2 and y = (1 − x2) + 5. Notice that the second graph has the sameshape, but is shifted 5 units higher.

What would happen if we subtracted 5 instead of adding 5? The graph of h(x) = (1− x2)− 5would have the same shape as the graph of f(x) = 1− x2 but would be shifted 5 units down. Weare subtracting 5 from the outputs of the function f to get the outputs of the function h. We couldwrite h(x) = f(x) − 5. Note that we could think of this as adding a negative constant instead ofsubtracting.

Horizontal shifts: Adding a Constant to the Input

Next, let’s consider the same function f(x) = 1 − x2, but let g(x) = f(x + 5) which equals1 − (x + 5)2. Here, we are adding 5 to the input of the function f instead of the output. In thiscase, the graph is shifted 5 units horizontally to the left. The graphs are shown in this case on thesame set of axes in Figure 4.

The shift to the left is probably the opposite of what you would expect, since the positivedirection on the x-axis is to the right. One way to convince yourself that left is the correctdirection is to consider a particular point. The point (0, 1) is the vertex of the graph of y = f(x).The vertex of the graph of g(x) will be (−5, 1), since g(−5) = 1− (−5 + 5)2 = 1, and g evaluatedat any other value of x will give a smaller output. So, the vertex of the graph of f gets moved 5units to the left to get the vertex the graph of g.

Vertical stretches and compressions: Multiplying the Output by a Constant

Next, consider the functions f(x) = 1 − x2 with g(x) = 3f(x) = 3(1 − x2). Here, we aremultiplying the outputs of the function f by 3. Thus, for each input x, the output g(x) is threetimes larger (in absolute value) than the output f(x). Graphically, this means the correspondingpoint on the graph of g is three times further from the x-axis than the point on the graph off . Points above the x-axis are three times higher, and points below the x-axis are three timeslower. Points on the graph of f which are on the x-axis have output 0, and since 3 · 0 = 0, thecorresponding points on the graph of g are still on the x-axis as well. As with vertical shifts,changing the output in this case also has an effect in the vertical direction, but in a different way.The graphs of y = f(x) and y = g(x) = 3f(x) are shown in Figure 5.


y = 1 x 2−

y = 1 x 5+( ) 2−

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-8 -6 -4 -2 0 2 4

x


Figure 4: Graph of y = 1 − x2 and y = 1 − (x + 5)2. Notice that the second graph is shiftedhorizontally 5 units to the left

y = 1 x 2−

y = 3 1 x 2−( )

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Figure 5: Graph of y = 1 − x2 and y = 3(1 − x2). Notice that the second graph is verticallystretched by a factor of 3.

What do you think would happen if you multiplied the output by (1/3)?

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Horizontal stretches and compressions: Multiplying the Input by a Constant

Finally, consider again f(x) = 1−x2 and let g(x) = f(3x) = 1−(3x)2. Here, we are multiplyingthe input of the function f by a factor of 3 to create the function g. The result is that the graphof f is compressed horizontally by a factor of 3. See Figure 6. Again, you might wonder why thisdoes not result in a horizontal stretch.

Consider a specific point on the graph of f , say (1, 0). To get the same output of 0 from thefunction g, we would let the input be 1/3, since g(1/3) = 1− (3 · 1/3)2 = 1− 12 = 0. So the point(1/3, 0) on the graph of g corresponds to the point (1, 0) on the graph of f . This point is a factorof 3 closer to the the y-axis.

y = 1 x 2−

y = 1 3x( ) 2−

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Figure 6: Graph of y = 1− x2 and y = 1− (3x)2.

To get a horizontal stretch, multiply the input by a number between 0 and 1. For example,the function h(x) = f(.5x) would result in a horizontal stretch by a factor of 2 of the graph of thefunction y = f(x).

One note on considering transformations from a visual standpoint. Remember that how thegraph of a function looks depends on the window that you use. We could have created the visualimpression of stretches and compressions without changing the function, but rather by changingthe window in which we graph the function. As we continue to look at transformations, we justneed to keep this in mind.

Summary of Shifts and Stretches

In summary, output changes for y = f(x) include:

• y = f(x) + c is the graph of y = f(x) shifted up c units (assuming c > 0).

• y = f(x)− c is the graph of y = f(x) shifted down c units (assuming c > 0).

• y = cf(x) is the graph of y = f(x) vertically stretched by a factor of c if c > 1.

• y = cf(x) is the graph of y = f(x) vertically compressed by a factor of 1/c if 0 < c < 1.


Input changes include:

• y = f(x + c) is the graph of y = f(x) shifted to the left c units (assuming c > 0).

• y = f(x− c) is the graph of y = f(x) shifted to the right c units (assuming c > 0).

• y = f(cx) is the graph of y = f(x) horizontally compressed by a factor of c if c > 1.

• y = f(cx) is the graph of y = f(x) horizontally stretched by a factor of 1/c if 0 < c < 1.

Application: Child Mortality

The graph in Figure 7 shows the Under Five Child Mortality Rate for three African countries,starting in 1960. The units for the dependent variable are in deaths per 1000 live births. The goodnews is child mortality is down in all three countries. All the graphs are concave up and decreasing.This would suggest an exponential curve. However, the graphs seem to be leveling off at mortalityrates significantly higher than zero. An exponential curve would level off at y = 0.Mortality Rate : Children Under Five

01002003004005006000 10 20 30 40 50Years from 1960

SierraLeoneMaliCongo,DRFigure 7: Mortality for Children Under Five for Three African Countries

We cannot know, of course, if the mortality rates will actually level off or not, and even if theydo, we cannot know at exactly what level. So, we will make an assumption that seems reasonableand go from there. A mortality rate of 200 seems a reasonable first assumption, at least for theCongo and Mali. Sierra Leone seems to be leveling off at a higher mortality rate, or it could be thatit will just take longer to get down to a rate of around 200. If this is a reasonable assumption foreach of the three graphs, then exponential models shifted 200 units up should give us a reasonablefit for all three. Recall that an exponential model is of the form y = abx. So, an exponential modelshifted 200 units up would be of the form y = abx + 200.

To test whether our assumption is reasonable, we can re-express the data, taking into accountthe shift we want to do. If we subtract 200 from all the mortality rates data values, this will havethe effect of shifting the mortality line graphs down 200 units. This will give values which “should”level off at 0 instead of 200, if our assumption is correct. We could then investigate whether anexponential model provides a good fit for the shifted data.

To find models, we could do exponential regression on the “shifted” data and then shift theresulting function up 200. Table 2 gives the shifted data for Sierra Leone and Mali.

For Sierra Leone, the exponential regression on the shifted data is y = 183(0.988)x with r =0.981. The final model for the original data would then by y = 183(0.988)x + 200. For Mali, thefinal model, using exponential regression and then shifting up by 200 is y = 316(0.944)x +200. For

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Years from 1960 0 10 20 30 35 40 41Sierra Leone 190 163 136 123 117 116 116Mali 317 191 95 54 43 33 31

Table 1: Shifted data for Sierra Leone and Mali (Mortality minus 200)

Mali, r = 0.999. These models are graphed along with the original data in Figure 9. Both modelsseem to give a very good fit, with the model for Mali even better than that for Sierra Leone.Mortality Rates: Mali and Sierra Leone

01002003004005006000 10 20 30 40 50Years from 1960

Figure 8: Mortality for Children Under Five with Shifted Exponential Models: Mali and SierraLeone

Could we have found even better models by making a different assumption about where themortality rates will level off? In the case of Mali, probably not. In the case of Sierra Leone, weprobably could get a better fit of the existing data by assuming a higher mortality rate as theleveling off point. In fact, by shifting the Sierra Leone data down 300 instead of 200, and followingthe same steps above, we get a model y = 91(0.956x) + 300 with correlation on the regression of0.994, versus the 0.981 we achieved previously. This may not seem like a huge improvement, butif you compare the scatterplot of the data with both models on your own, you should see that thesecond model does seem to give a better fit.

One final word of caution. As with all empirical models, we must be aware that the predictionsthe model provides are subject to error, and that we should be less confident of the predictionsthe further into the future we go. The model for Mali may accurately predict what happens inthe early 21st century or it may not. However, if mortality rates decrease more rapidly than themodel predicts (or if they reverse the trend and start to increase), we will at least have a basis forcomparison. We might then ask “what, if anything, has changed in Mali that might explain thedifference between our predictions and reality?”

Reading Questions for Transformations of Functions

1. Match each of the following function transformation operations with the correct graphicaltransformation.A. f(x + c) (i) vertical stretchB. cf(x) (ii) horizontal shiftC. f(x) + c (iii) horizontal stretchD. f(cx) (iv) vertical shift

2. Given the accompanying graph of y = f(x), match the following function transformations tothe following graphs.115

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-2 -1 0 1 2 3 4

(a) y = f(2x) (b) y = 2f(x) (c) y = f(12x)

(d) y = f(x) + 2 (e) y = f(x− 2) (f) y = f(x + 2)

(i)-3

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115Exercise adapted from Andersen and Swanson

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3. Suppose that t(c) represents the function in which c is the number of credits you are takingthis semester and t(c) is the cost of tuition. Suppose a change was made in the cost of tuitionsuch that 100 + t(c) describes the new tuition function. This new function describes whichof the following situations?

(a) The number of credit hours has increased by 100.(b) The cost per credit has increased by $100.(c) Tuition has increased by 100%.(d) The total cost of tuition has increased by $100 (e.g. a technology fee was added).

4. Which of the following type(s) of transformations will always result in all the points on thegraph moving. (Circle all that apply.)

(a) adding a constant to the output(b) multiplying the output by a constant(c) adding a constant to the input(d) multiplying the input by a constant

5. Let y = f(x) be given below. Sketch a graph and give the domain and range for each ofthe following functions. Be sure to label the coordinates of the two endpoints and the two“corner” points on each graph.

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(a) y = f(x)− 1(b) y = f(x− 1)(c) y = 2f(x)(d) y = f(x) + 2(e) y = 1

2f(x)

(f) y = f(

12x

)

(g) y = f(2x)

6. Suppose that y = t(c) represents the function in which c is the number of credits you aretaking this semester and y = t(c) is the cost of tuition. If the cost per credit increases by 10%(which means the cost of tuition is 110% times last year’s cost), what function represents thenew cost of tuition?

(a) y = t(1.1c) (b) y = t(c + 1.1) (c) y = t(c) + 1.1 (d) y = 1.1t(c)


7. Suppose a navy submarine was exercising tactics by beginning at sea level, diving to −300feet in 1

2 hour, then climbing to −100 feet in 12 hour. It then dove to −500 feet in 1 hour

where it rested for 2 hours. Let y = s(t) be the function where t is time and s(t) is thedistance of the submarine from the surface.

(a) Sketch a graph of y = s(t). Let the horizontal axis represent sea level.

(b) Sketch a graph of y = s(12 t). What did the submarine do differently for this graph?

8. Suppose a person left an office, walked slowly down a hall at a constant rate, stopped to talkto someone for a few seconds, and ran back to the office when the phone rang. Sketch a graphof this event where time is the input and the distance from the person to the office door isthe output.

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Transformations of Functions: Activities and Class Exercises

1. Changing units and Transformations.

(a) In the Chapter 2 activities, we considered the function w(h) = 1.4h − 20 which is alinear model for the average weight, in pounds, of a child whose height (or length) isgiven in inches. As part of that activity, we also considered the functions:

i. y = 0.45w(h) which gives the weight in kilograms as a function of the height ininches. Let’s denote this function k(h). What kind of transformation, applied tow(h), gives the function k(h)?

ii. y = w( c2.54) which gives the weight in pounds as a function of the height c in

centimeters. Let’s denote this function m(c). What kind of transformation, appliedto w(h), gives the function m(c).

iii. y = .45w( c2.54) which gives the weight in kilograms as a function of the height in

centimeters. Let’s denote this function G(c). Describe how G(c) is created fromw(h) via transformations.

(b) In Chapter Eight, one of the activities involved creating a power function to model babyweights, using the data below, as a function of the time in months.

Age (months) 3 6 9 12 15 18 21 24Weight (lb) 12.3 16.2 19.1 21.1 22.8 24.0 25.0 25.8

The power regression function for these data is w(x) = 8.48x.36, where w is the weightin pounds and x is the age in months.

i. Suppose you want a function to model baby weights as a function of the time inyears instead of months. Use a function transformation of the power regressionfunction w to create this new function.

ii. Suppose you want a function to model baby weights in kilograms instead of pounds.Use a function transformation on w to create this function.

iii. Suppose you want to model baby weights as a function of the time in months, butstarting at conception instead of at birth. Assuming a nine month pregnancy, usethe function w to create this new function.

(c) Earlier in this chapter, we created shifted exponential models for child mortality ratesfor Mali and Sierra Leone where the independent variable was the time in years from1960. The model for Mali was y = 316(.944)x + 200.

i. Use a transformation to transform this model into a model which gives the childmortality rate for Mali as a function of the actual year instead of the year from1960.

ii. To check your model from part (i), graph your model together with a scatterplot ofthe data, but using the actual year as the independent variable in your scatterplotinstead of the year from 1960.

iii. We could have used the actual year as our independent variable instead of theyear from 1960 when we created our child mortality model for Mali. Discuss whatdifference this would have made. Discuss any advantages or disadvantages of usingthe actual year versus using the year from 1960?


2. Child Mortality and Literacy.

In Figure 1 of this chapter, the relationship between illiteracy and childhood mortality forfemales as measured by survival rates to age 10 was depicted in a scatterplot. The data onwhich this graph was based is given below.

Pct. Newborn Pct. NewbornIlliteracy Fem. Surviving Illiteracy Fem. Surviving

Country Rate to Age 10 Country Rate to Age 10Barbados 0.3 98.6 Panama 8.8 97.6Guyana 1.9 93.1 Jamaica 9.3 98.4Uruguay 2.0 98.5 Ecuador 10.1 96.4Trinidad & Tobago 2.4 98.6 Mexico 10.9 97.4Argentina 3.2 97.8 Brazil 13.2 95.6Cuba 3.4 99.1 Peru 14.8 94.9Bahamas, The 3.7 98.5 Dominican Rep. 16.3 95.1Costa Rica 4.4 98.4 Bolivia 20.8 91.3Chile 4.4 98.9 El Salvador 24.0 96.4Belize 6.9 96.5 Honduras 25.0 95.2Paraguay 7.8 96.7 Nicaragua 33.3 95.6Venezuela 8.0 97.7 Guatemala 38.9 94.4Colombia 8.4 97.7 Haiti 52.2 89.3

(a) As in the Childhood Mortality example, use the following steps to create a shiftedexponential model for these data.

i. Assume the scatterplot will level off at y = 88.ii. Input the data in a calculator or appropriate software package.iii. Add a column re-expressing the dependent variable by subtracting 88 from each

survival rate.iv. Find the exponential regression model with the illiteracy rate as the independent

variable and the re-expressed survival rates as the dependent variable.v. Shift your exponential model up by 88 to create a model for the original data.

(b) Graph your shifted exponential model together with the original data. How well doesyour model fit?

3. Child Mortality in Albania. The table below gives data on child mortality in Albania.A graph is also provided. The scatterplot is decreasing, and the trend seems to be levelingoff towards zero over time. The general shape is concave up. All these indicate that anexponential model with base between 0 and 1 might be appropriate.

(a) Find the exponential regression model for this data, and graph the data and the modeltogether. How confident are you that the model will make reasonable predictions forchild mortality for years after 2000?

(b) As in the application on child mortality in this chapter, create a shifted exponentialmodel for the Albanian data, assuming that the mortality rate will level off at y = 20.Give the correlation coefficient that you get from the exponential regression on theshifted data, along with the equation for the shifted exponential model.

(c) Plot this shifted model together with the data. Does this model seems to fit the databetter than your first model?

204

Year Childfrom 1960 Mortality

0 15110 8220 5730 4535 3540 31

Table 2: Child Mortality as a function of Year since 1960Child Mortality Albania (Deaths per 1000 live births)020406080

1001201401601950 1960 1970 1980 1990 2000 2010

Figure 9: Child Mortality Rates for Albania over time

(d) Using both of your models, predict Child mortality in Albania for the year 2020 and2050. Which predictions are you most confident in and why?

4. Cost of Computer Storage. In 1956, one IBM hard drive could store 5 megabytes ofinformation at a cost of $50,000. In 1981, Creative Computing magazine predicted “the costfor 128 kilobytes of memory will fall below US $100 in the near future.” 116 This wouldrepresent about $800 per megabyte. The graph below shows an exponential model, y =15.85(.487x), approximating the cost for 1 gigabyte (1000 megabytes) of storage where xrepresents the year from January 1st, 2000. Note that this model predicts one gigabyte cost$15.85 on January 1st of 2000, and that eventually storage costs will become close to $0 pergigabyte. The equation for this graph is y = 15.85(.487x).

116http://www.alts.net/ns1625/winchest.html accessed on December 30th, 2007


y = 15.85 0.72 x−( )exp

0

4

8

12

16

20

x0 1 2 3 4


(a) The graph is decreasing and concave up. Carefully explain what both of these mean forthe cost of computer storage.

(b) Suppose one gigabyte started out at a price of $15.85 in 1998 rather than 2000. Assumingthe original change in price remains the same (in other words, that the exact shape ofthe graph is not changed), identify the transformation this implies (vertical/horizontalstretch/shift, etc.) and find the equation for this new function.

(c) Suppose computer storage started out in 2000 at the higher price of $20. Assuming theoriginal change in price remains the same, what transformation is this and what is theequation of this new function?

(d) Suppose both things in parts (b) and (c) occurred, so that computer storage started outin 2000 at a price of $20 per gigabyte.

(e) Assuming the original rate of change of price remains the same, give the equation of thenew function.

(f) Suppose we write the original function in function notation as f(x) = 15.85(.487x).Also, suppose that the price of computer storage had instead decreased according to thefunction y = f(1.5x). What type of transformation is this?

(g) According to the original model, what should the price of a gigabyte of storage be today,and when will the price reach 1 cent per gigabyte?

5. Newton’s Law of Cooling. Suppose it’s a cold winter morning, and you have madeyourself a nice cup of hot chocolate. You set it down on a table when the phone rings andsubsequently forget about it. It will, of course, cool off over time, eventually getting down toroom temperature.

(a) By hand, draw a reasonable graph of the temperature of the hot chocolate versus time.Include scales on both the horizontal and vertical axis. You might use minutes for thehorizontal axis and degrees Fahrenheit (or Celsius) on the vertical. Consider carefullythe concavity of the graph.

(b) What family of functions would seem to be most appropriate for modeling your graphfrom part (a)? You might want to consider a transformation of one of the basic familieswe have considered.

(c) Assume that your hot chocolate was at 150◦ Fahrenheit when you set it on the table,and that the room containing the table is at 70◦ Fahrenheit. If you were to use atransformed exponential function of the form y = abt + c to model the temperature y

206

as a function of the time t, what could you say about the parameters a, b, and c basedon this information? (Hint: You will not be able to exactly determine the value of oneof these parameters, but should be able to provide partial information for it)

(d) Assume the same information as in part (c), and in addition, suppose that the tempera-ture of the hot chocolate was 120◦ degrees after 10 minutes. This is enough informationfor you to determine all three parameters a, b, and c. Find all three parameters.

(e) Later that day, you lift some weights in the gym. Afterwards, you are hot and decide tohave a cold drink. You buy a can of pop which has a temperature of 45◦ Fahrenheit, andopen it, at which time the fire alarm goes off and you have to leave the building. Youleave your pop sitting in the weight room, which is at a temperature of 78◦ Fahrenheit.

i. How would a graph of the temperature of the pop versus time be different than inthe case of the hot chocolate? How would it be similar? Consider both the concavityand whether the graph is increasing or decreasing.

ii. You could use a transformed exponential function of the form y = abt + c to modelthe temperature y of the pop over time. Use the information you have to determinethe parameters a and c. (Hint: a should be negative)

iii. Do you have enough information to find b for this warming pop example?iv. Assume b = .8, and graph the equation y = abt + c using the parameters a and c

you have already found for the pop. You should get an increasing and concave downgraph. Graphically, you have an exponential function but “flipped upside-down”,since it is concave down instead of concave up. This type of transformation is calleda reflection over the x-axis. Symbolically, to reflect the graph of a function f(x)over the x-axis, which holds the points on the axis fixed and flips everything elseover the axis, you would graph y = −f(x).

10. Other Families of Functions

How much space do you need?

As of January of 2004, there were about 6.2 billion people in the world. The total land areaof the world is about 57.4 million square miles, with 5.1 million of that in the Antarctic. Anotherroughly 7 million square miles is under water of one form or another (glaciers, lakes, rivers, etc.),leaving about 45 million square miles of “exposed land.”

Of course, a significant portion of the Earth’s exposed land is uninhabitable or cannot be usedfor any agricultural purpose due to arid or mountainous terrain, or year-round frigid conditions.In fact, only around 13 million square miles is arable. Currently over half of this area is not undercultivation.

How much land does this leave for each of us to “live on?”

If we measure population density in persons per square mile, using the total amount of exposedland, we get

6.2× 109

45× 106=

6.445

× 103 ≈ 142 people per square mile

This would give each person about 200,000 square feet. This is about 4 football fields. Aboutone and one-half of these football fields would be arable land.

Is this enough land for a person to live on?

Keep in mind that not only do we need actual living space, we need land to provide many of theother necessities of life, most notably food, but also energy, water, clothing, material for shelter,etc.

Whether this 200,000 square feet is enough is a complicated question. However, it is reasonableto assume that, even with improved agricultural techniques and more advanced technology, thereis some upper limit on how many people the earth can reasonably support. This implies that if wewish to model world population in the long term, we want a model that “levels off” at some point.In mathematical terms, we want a function which has a horizontal asymptote y = M , where M isthe positive value representing the upper limit on human population. What kinds of functions canhave such a horizontal asymptote?

Families of Functions So Far

Although we have considered a variety of families of functions and have used transformationsto adjust the basic families to create more and more accurate models, all of our functions havebeen really very simple functions in two important ways. First, with only one exception, all ofour symbolic functions have been either always increasing or always decreasing. We have not useda symbolic function which “changes direction,” except for power functions where the power is aneven integer.117 Mathematicians call functions with no change from increasing to decreasing orvice versa monotonic. Second, with one exception, all of our symbolic functions have been eitheralways concave up or always concave down. Mathematicians would say that such functions haveno inflection points, that is, points where the concavity of the function changes from concavedown to concave up, or vice versa. The only exceptions to this were the power functions where thepower is an odd integer (eg. y = x3). These have an inflection point where x = 0.

We can create increasing functions which “level off” at a positive horizontal asymptote usingtransformations of exponential functions or power functions with negative exponents. In each of

117Such functions change direction at x = 0, like y = x2.

207

208

World Population in Billions

0

2

4

6

8

10

1950 1970 1990 2010 2030 2050

Figure 1: World Population from 1950, projected to 2050

these cases, the model would be monotonic increasing. Can we create models for situations likeworld population, shown in Figure 1? In both of these cases, the graphs of the data show a changein concavity as well as a leveling off over time.

What about data which shows both increases and decreases, and other more complicated be-havior, like Figure 2 of Chapter Two, which shows total world energy production?

In this chapter, we will look at a few other families of functions that we can use to modeldata that has more complicated behavior. This will certainly not be a comprehensive treatment.However, it will give you a few more families to use as tools. The good news is that there areseveral standard models which we can use fairly easily. The trade off is that the more complicatedthe model, the fewer will be the situations where it gets used. Also, we need to be careful not todeceive ourselves that a more complicated model always means a better model.

Logistic Growth Models

The family of logistic functions can provide reasonable models for quantities that increase overtime, but where there is or is assumed to be a maximum value beyond which the quantity will notgo. The basic shape of logistic curves is an elongated “S” shape, similar to the graph of urbanpopulation in Botswana.

The Number e

There are several ways to express the symbolic form for logistic functions. The most usualexpression involves the number e, which we introduced in Chapter Six. Also, recall that when weintroduced logarithms, we said

y = logb x means the same as by = x

and that when b = 10, we simply write log x. If b = e ≈ 2.71828, we write lnx instead of loge x.Now, the symbolic form for a logistic function L(t) is

L(t) =M

1 + be−rt

We are using t to denote the independent variable since we most often use logistic functionsto model data where the independent variable is time. In this expression, M , b, and r are theparameters. These parameters have the following interpretations:

10. Other Families of Functions 209

(1) M represents the limiting value of the function. For logistic functions which are increasing,118

M is the upper limit of growth, beyond which the function will not increase.

(2) r represents the approximate percentage growth rate in L(t) when L(t) is much smaller thanM .

(3) b = ML(0) − 1, where L(0) represents the initial amount for the quantity L.

(4) Increasing logistic growth curves have an inflection point, where the concavity changes fromconcave up to concave down, at the point where L(t) = 1

2M . This occurs when t = ln(b)r .

Example 1. It is possible to make yogurt yourself, using commercial yogurt (as long as itcontains live yogurt cultures) and milk. Simply put a few tablespoons of yogurt in a jar of warmmilk (about 105 degrees Fahrenheit) and maintain the temperature for about 4 hours or so. Theyogurt organisms will eat the milk and you will end up with a jarful of yogurt (and no milk), anda small amount of waste product (called whey, as in what Little Miss Muffet ate with her curds).

Suppose we let L(t) be the amount of yogurt measured in ounces in a 32 ounce container attime t, measured in hours, during a yogurt making process. Assume the initial amount of yogurtis 1 ounce and that the percent increase per hour in the amount of yogurt for small amounts ofyogurt is r = 1.8 which represent a 180% growth rate. Assuming L is a logistic growth function,find the symbolic representation for L. When does the model predict that there will be 31.5 ouncesof yogurt?

Solution: Under the assumptions given, we have r = 1.8, L(0) = 1 ounce, and M = 32 ounces.We can find b using (3) above. We have b = M

L(0) − 1 = 321 − 1 = 31. So, our logistic model is

L(t) =32

1 + 31e−1.8t

To find out when we get 31.5 ounces of yogurt, one way to proceed would be to graph y = L(t)and then estimate the t-value when we reach y = 31.5. Try this on your own graphing technologyand you should get about 4.2 hours. The graph is shown in Figure 2.Ounces of Yogurt

051015202530350 1 2 3 4 5 6Time in HoursFigure 2: The Growth of Tomorrow’s Breakfast

Alternatively, we could solve the equation

118logistic functions can be decreasing as well

210

31.5 =32

1 + 31e−1.8t

for t. We can do this with some algebra and logarithms. We have,

31.5 =32

1 + 31e−1.8t

1 + 31e−1.8t =32

31.5

31e−1.8t =32

31.5− 1 ≈ .015873

e−1.8t ≈ .01587331

≈ .000512

Using the natural logarithm, we can change this to exponential form, getting ln .000512 = −1.8t or

t ≈ ln .000512−1.8

≈ 4.21

very close to the estimate given above.

Explanation of the Parameters

Notice in Figure 2 that the growth curve does have an inflection point at about L = 16, whichis half of the limiting value of M = 32, illustrating property (4) above. This is where the yogurtpopulation is growing most rapidly. Remember that when a function is concave up, this meansthat the rate of change of the function is increasing, and when a function is concave down, the rateof change is decreasing. Logistic curves, at least those where L(0) < 1

2M , will have increasing ratesof change until the quantity L reaches half of the limiting value, after which the rate of growth willdecrease.

Next, let’s explain properties (1), (2), and (3). You will be asked to consider property (4) inthe Reading Questions. Property (3), which says b = M

L(0) − 1, is perhaps the easiest. If we lett = 0 in our basic equation L(t) = M

1+be−rt , we get

L(0) =M

1 + be−r·0 =M

1 + be0=

M

1 + b

1 + b =M

L(0)

b =M

L(0)− 1

Next, consider the parameter M , the limiting value for the function L, or the value at whichL levels off. M will be the limiting value if the values of L(t) get closer and closer to M as t getslarge. We need to consider the expression e−rt which is part of the function L(t).

Recall that negative exponents mean “take the reciprocal.” So, e−rt = 1/ert. What happensto this expression if we evaluate it at large values of t? Well, if t is very large, than rt will still belarge, since r will be positive. Even if r is only .01, for example, if we pick t larger than 10,000,then rt will be larger than 100. If we take the number e to a large power, since e > 1, we get aneven larger number. In fact, if rt = 100, ert will be about 2.7× 1023, a huge number. So, for largevalues of t, e−rt = 1/ert will be 1 divided by a large number, and so will be very close to 0. Now,looking at the whole function L(t) = M

1+be−rt , if e−rt is approximately 0, then

L(t) =M

1 + be−rt≈ M

1 + b · 0 =M

1 + 0= M

10. Other Families of Functions 211Yogurt Growth: Logistic and Exponential05101520

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 LogisticExponentialFigure 3: Logistic and exponential growth models for yogurt growth

Time Logistic Exponential0.0 1.00 1.000.1 1.19 1.200.2 1.41 1.430.3 1.68 1.720.4 1.99 2.050.5 2.35 2.460.6 2.78 2.940.7 3.27 3.530.8 3.83 4.220.9 4.49 5.051.0 5.23 6.051.5 10.38 14.882.0 17.33 36.60

Table 1: Numerical comparison of logistic and exponential yogurt growth

So, M is the limiting value which L approaches as t gets large. Also, note that if b > 0, then so isbe−rt. This implies that

M

1 + be−rt<

M

1= M

since the fractions have the same numerator and the denominator of the second fraction is smallerthan the denominator of the first.

How large t must be for L to be close to M depends on r and b. If r is very small, this indicatesa slow growth rate and so it will take longer for L to approach M . Similarly, the smaller b is, thelarger t needs to be for L to be close to M .

Finally, let’s consider the role of r in the logistic model. A complete mathematical explanationof why r is approximately equal to the percentage growth rate when L is much smaller than Mrequires calculus. However, we can get an idea of how the logistic and exponential models comparenumerically and graphically. Table 1 shows values for the logistic model of yogurt growth given inExample 1, along with the exponential model E(t) = e1.8t which has the same initial amount ofE(0) = 1 and the same percentage growth rate. At least over most of the first hour, the exponentialmodel and the logistic model are very close. A graph of the two models over the first 1.5 hours isshown in Figure 3.

212

Application: World Population Growth

One of the main challenges in using logistic models is that, unlike the situation with our yogurtexample, we rarely know what the maximum value M is ahead of time. Finding L(0) is no problem,and we might be able to estimate r if we can be reasonably sure that the data we have is for valuesmuch less than whatever the maximum value is. However, without M we cannot directly find b andwill not be able to get a model, at least in the same way we found our yogurt model in Example 1.

However, in cases where it looks like the quantity being modelled has already reached a pointwhere the rate of growth is decreasing, then we can use property (4) to make an assumption onwhat M to use. In Figure 1, although there is not a great deal of “curve” to the graph, it looks asif the graph is steepest around 1990 or just before, and this is where the graph is changing fromconcave up to concave down. In fact, Table 2 shows the world population and the rates of growthfor years around 1990. The rates of growth are given in number of people per year, and so theyear which sees the biggest additional number of people is the year where the rate of growth is thelargest.

Year Population Percent Growth Population Increase1980 4,454,269,203 1.69 75,864,5641981 4,530,133,767 1.75 80,105,0081982 4,610,238,775 1.73 80,253,7641983 4,690,492,539 1.68 79,312,0071984 4,769,804,546 1.68 80,596,5051985 4,850,401,051 1.68 82,324,4171986 4,932,725,468 1.71 85,142,8121987 5,017,868,280 1.69 85,667,3321988 5,103,535,612 1.66 85,671,9961989 5,189,207,608 1.66 86,677,6811990 5,275,885,289 1.58 83,940,3511991 5,359,825,640 1.55 83,939,7111992 5,443,765,351 1.48 81,404,0541993 5,525,169,405 1.44 80,191,4341994 5,605,360,839 1.43 80,626,2571995 5,685,987,096 1.38 79,173,661

Table 2: World Population, 1980-1995, with percentage growth rates and growth rates in personsper year

The table confirms what we saw from the graph. The largest rate of change occurred in 1989,when the population increased by over 86 million people. If the world population is growingaccording to a logistic curve, then the population in 1989 should be about half of the limitingpopulation. Under this assumption, the limiting population would be M = 2 · 5, 189, 207, 608 orabout 10.4 billion people.

We can now create a logistic model for world population. We will use M = 10.4 and let t = 0represent 1950. The population in 1950 was 2.56 billion so we let L(0) = 2.56. This gives us avalue for b of 10.4/2.56− 1 = 3.0625. To complete our model, we need to find r. We could do thisby using the parameters we have and one data point to create an equation. Which data point is asomewhat arbitrary choice, and we will use the population in 1980 which is 4.45 billion. The year1980 corresponds to t = 30. From the basic model L(t) = M/(1 + be−rt), we get

L(30) = 4.45 =10.4

1 + 3.0625e−30r

1 + 3.0625e−30r =10.44.45

≈ 2.337


3.0625e−30r ≈ 1.337

e−30r ≈ 1.3373.0625

≈ 0.43657

−30r ≈ ln 0.43657 ≈ −0.8288

r ≈ −0.8288−30

≈ .027627

This gives a logistic model for world population from 1950 of

L(t) =10.4

1 + 3.0625e−0.027627t

This curve is plotted along with a few of the actual data points as a scatter plot in Figure 4.World Population with Logistic Growth Model02468101950 1970 1990 2010 2030 2050

Figure 4: World Population with Logistic Model

The model follows the data fairly well over the first 40 years or so, but then underestimates thevalues after that. This may seem a little disappointing. However, we should keep in mind that allthe values after 2003 are estimates, and in fact, would have been created by some other model.119

In the Reading Questions for this chapter, you will be asked to create another logistic model forthis data, which will hopefully give a better fit than this one.

Hubbert Curves

Figure 5 shows a graph depicting the total oil production of Norway, a significant exporter ofpetroleum.120

As noted previously, Hubbert was a geologist who used bell-shaped curves that have subsequentlycome to bear his name to approximate oil production data. Symbolically, Hubbert curves are ofthe form

H(t) =Mbre−rt

(1 + be−rt)2

119This other model may be a different logistic model, but more likely, is a more complicated model which alsotakes into account the actual and projected percentage growth rates

120Graph is from http://www.hubbertpeak.com/blanchard/

214

Figure 5: Norwegian Oil Production in Barrels per Day

where the parameters M , b, and r are the same as those in the logistic family of functions.121 Withrespect to Hubbert curves

• M represents the total area under the curve.

• The peak occurs at t = ln(b)/r.

• The maximum height at the peak is equal to Mr/4.

If we wanted to find a Hubbert curve to model the Norwegian oil production data, we couldfirst estimate the total area under the curve, which would be an estimate for M . To do this, let’sestimate half of the area and then multiply by two. Looking at the graph, the peak year seems tobe about 1993. As a rough estimate, we could say the area under the curve up to this year is aboutthe area of the triangle with two vertices along the x-axis at the years 1980 and 1993, and the thirdvertex at the peak, which is at the year 1993 with a height of about 2,000,000 b/d or barrels perday. The base of this triangle has length 13 and the height is about 2,000,000. So, using the 1

2bhformula for the area of a triangle, the area is about

12· 13 · 2, 000, 000

which is half the area under the curve. So, the whole area is about 13 · 2, 000, 000 or about26, 000, 000 = 2.6× 107. So, we can let M = 2.6× 107.

The maximum height of the curve is about 2 million or 2 · 106. This would equal Mr/4 =(2.6 · 107)r/4. So,

2 · 106 = (2.6 · 107)r/4

4 · 22.6 · 10

= r

so that r = .3077.Now, ln(b)/r is the t-value of the peak, in this case 1993. So ln(b) = 1993r = 1993 · .3077 =

613.25. Solving for b, we get that b = e613.25, which is about 2.135 ·10266, a truly huge number. Wecould still use this for our model, although it might cause problems for a TI calculator. If we’d like,we could simplify things by letting t = 0 denote 1970 so that the peak would occur when t = 23.

121For those of you who do know calculus, take the derivative of the formula which gives the general form of logisticfunctions, and you should get the expression above for Hubbert curves


In this case, ln(b) = 23r = 23 · (.3077) = 7.077, and b = e7.077 ≈ 1184, a much more manageablenumber. This gives a model

H(t) =2.6 · 107 · 1184 · (.3077)e−.3077t

(1 + 1184e−.3077t)2=

9.472 · 1019e−.3077t

(1 + 1184e−.32077t)2

where t represents the year from 1970.Table 3 gives some values of this Hubbert curve model. You can compare these with estimated

values from the graph to see how closely it fits.

Year Hubbert1980 141,3491985 579,5561990 1,628,1931993 2,000,0002010 42,3172020 1,970

Table 3: Numerical values of Hubbert curve model for Norway

Piecewise Functions

Some functions given in symbolic form have different rules for different parts of their domain.For example, the function used for determining income tax involves different symbolic formulas fordifferent income levels. Such functions are called piecewise functions. When determining theoutput for a given input, it is important to make sure you are using the correct rule. For example,let

f(x) =

{−x if x < 0x2 otherwise.

Then f(−4) = −(−4) = 4 since −4 < 0, while f(0) = 02 = 0 and f(4) = 42 = 16. The decisionwhether to use the rule “find the opposite of the input” or the rule “square the input” depends onwhether your input is less than zero.

Figure 6 is the graph of a piecewise function. Notice the abrupt change at x = 3. Graphs ofpiecewise functions often have an abrupt change at the point where the rule changes.

0

2

4

6

8

10

12

14

16

-4 -2 0 2 4 6x

(3,9)

Figure 6: The graph of a piecewise function.

216

The symbolic representation of the function shown in Figure 6 is

f(x) =

{x2 if x < 3−2x + 15 otherwise.

Many modeling functions are piecewise functions. For example, phone companies often offerplans which charge customers a flat rate for so many minutes, and then a per minute charge if thecustomer uses more than the limit. For example, Verizon has offered a cell phone plan that costs$39.99 per month and includes 450 anytime minutes. The cost per minute for calls going over the450 minute limit is $.45 per minute.122 This verbal description can be represented in the followingsymbolic form, where m is the number of minutes used in a month, and C(m) is the associate cost.

C(m) =

{39.99 if 0 ≤ m ≤ 45039.99 + 0.45(m− 450) if m > 450.

So, for example, C(300) = 39.99, and C(500) = 39.99 + .45(500− 450) = 62.49.

Example 2.

As another example, consider the Dakota Wesleyan University tuition cost function given ver-bally at the beginning of Chapter Four. Can we find a symbolic representation for this function?

Solution: We will give two symbolic functions, one exact and one approximate. First, we givethe function in numerical form in Table 4. We also included the differences between successive costvalues. Recall that if the difference in outputs for each unit increase in the input is constant, thenthe function is linear with the constant difference representing the slope. In this case, if portionsof the table show a linear pattern, we can form a piecewise functions with those pieces representedby a linear formula.

Hrs. Cost Diff. Hrs. Cost Diff.1 375 11 7725 16252 750 375 12 8750 10253 1125 375 13 8750 04 1500 375 14 8750 05 1875 375 15 8750 06 2525 650 16 8750 07 3150 625 17 9125 3758 3850 700 18 9500 3759 4475 625 19 9875 37510 6100 1625 20 10250 375

Table 4: DWU Tuition Costs for the Fall of 2007

From the table, we see that there is a linear pattern for credit hours between 1 and 5 inclusive, from9 to 11 inclusive, from 12 to 16 inclusive, and over 17. The pattern from 6 to 9 is approximatelybut not exactly linear. A piecewise function which gives the exact values for tuition T (c) for agiven number of credit hours c is given by:

122Plan details acquired at www.pricegrabber.com on August 17th, 2007


T1(c) =

375c if 0 ≤ m ≤ 52525 if m = 63150 if m = 73850 if m = 84475 + 1625(c− 9) if 9 ≤ m ≤ 118750 if 12 ≤ m ≤ 168750 + 375(c− 16) if 16 < m

Notice we have four linear pieces with slopes 375, 1625, 0, and 375 respectively. Unfortunately,we do still have a complicated function with 7 different pieces. We could make this simpler byapproximating the tuition costs for the credit hours between 6 and 8. One way to do this wouldbe to find the linear equation through the points (6, 2525) and (8, 3850). The slope is

m =3850− 2525

8− 6=

13252

= 662.5.

To find b, we solve 2525 = 662.5 · 6 + b for b to get b = −1450. If we use this formula for c valuesbetween 6 and 8, we get a second function with 5 different linear pieces.

T2(c) =

375c if 0 ≤ m ≤ 5662.5c-1450 if 6 ≤ m ≤ 84475 + 1625(c-9) if 9 ≤ m ≤ 118750 if 12 ≤ m ≤ 168750 + 375(c-16) if 16 < m

The only difference between these two functions will be that T1(7) = 3150 and T2(7) = 3187.50.Perhaps we could suggest that DWU make a slight modification of their tuition structure for nextyear. Also, note that the two of the linear formulas are not in y = mx + b form, but could besimplified to this form. We have

4475 + 1625(c− 9) = 4475 + 1625c− 14625 = 1625c− 101508750 + 375(c− 16) = 8750 + 375c− 6000 = 375c + 2750

Reading Questions for Other Families of Functions

1. Let

f(x) =

{4x if x ≤ 0x2 − 1 if x > 0

Compute each of the following.

(a) f(−2)(b) f(0)(c) f(3)

2. Let

g(x) =

2x if x ≤ 0x2 + 1 if 0 < x < 42x + 9 if x ≥ 4

Compute each of the following.

(a) g(−2)(b) g(0)(c) g(3)(d) g(4)(e) g(6)

3. In this chapter, we created a logistic model for world population, and commented that themodel seemed to underestimate the actual population values after 1990.

(a) Create a logistic model for world population, letting t = 0 correspond to 1970, but uselogistic regression instead of the technique in the text.

(b) Graph your model together with the data given in the appendices, starting in 1950, tosee how well the model fits. You may use an electronic file with the data. Otherwise,type in the data using only the years 1950, 1955, 1960, etc.

(c) According to this model, what will the eventual upper limit on human population be?How well does this model fit the data for the years 2000 through 2050.

4. Give an example (other than the one in the reading) of a piecewise function which is amodeling function.

5. In the reading, we looked at the cost of a particular cell phone plan from Verizon. AlltelWireless had a competing plan for $49.99 per month for 1000 minutes, plus 40 cents perminute over the 1000 minute limit.

(a) Find a symbolic representation for the piecewise function giving the cost of the Alltelplan as a function of the number of minutes used.

(b) If we let m represent the number of minutes, for which values of m does the Verizonplan cost less than the Alltel plan?

(c) If you want to be sure your monthly cost is below $80, how many minutes can you useunder each of these plans?

6. In the text, we created a function T2(c) which essentially gave the tuition cost at DWU forthe fall of 2007 as a function of the credit hours c.

(a) Suppose for the fall of 2008, DWU wants to increase tuition across the board by 4%over the tuition costs given by T2. Give a piecewise function in symbolic form whichdoes this.

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(b) Suppose for the fall of 2008, DWU wants to create a more simplified tuition structurewith only 3 linear pieces. The function should satisfy the following conditions:

• A student taking 0 credits is charged $0.• A student taking 12 to 16 credits is charged $9000.• Credits over the 16 hour limit are charged at $400 per hour.

Create a piecewise function with 3 linear pieces that meet these 3 conditions.

7. Suppose that H(t) = 120,000e−0.05t

(1+3.5e−0.05t)2is a Hubbert curve model for annual U.S. oil production,

where t = 0 represents 1950. The units for H(t) are in thousands of barrels per day.

(a) Graph y = H(t). Figure 1 of Chapter Three shows a scatterplot of U.S. oil production.How does the graph of H compare with this scatterplot?

(b) In which year does H show a peak?

(c) Use H to estimate the total U.S. oil production in 1950, 2000, 2020, and 2050.

(d) Comparing the symbolic representation for H with the general form for Hubbert curvesgiven in the text, we can see that for this curve, b = 3.5 and r = .05. Use these valuesand the general form to find M for this curve.

(e) In 2000, the U.S. consumed about 16,000 thousands of barrels of oil per day. In part(c), you estimated how much oil the U.S. produced in 2000. Based on this information,what percent of its oil needs did the U.S. have to import in 2000?

(f) The M that you found in part (d) represents the estimated total amount of oil thathas been and will ever be produced by the U.S. The peak year you found in part (b)represents the year in which half of this production would have already occurred. Thismeans that after that year, we only have half of our value of M left to use. Let’s beoptimistic and suppose that we still had half of M left to use in the year 2000. Assumingthat the U.S. consumes 16,000 thousands of barrels per day from 2000 on, and is ableto access this much oil every year from its own production sources, how long would itbe before the U.S. used up all its oil?

8. The graph below shows the cumulative number of AIDS cases reported for Zimbabwe, acountry in southern Africa, for each year. The values represent the total number of casesreported up to and including that year.New AIDS Cases - Zimbabwe

010000200003000040000500006000070000800001984 1986 1988 1990 1992 1994 1996 1998 2000

(a) Explain why a logistic model might be appropriate to model this data.

(b) The table below gives the data on which the graph above is based. Based on the graphand the table, estimate the maximum number of cases for a logistic model of these data.

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Year 1986 ’87 ’88 ’89 ’90 ’91 ’92 ’93 ’94 ’95 ’96 ’97 ’98Cases 0 119 321 1632 5994 10551 18731 27905 38552 51908 63937 70669 74782

(c) Find a logistic model for these data, and graph both the model and the data. How welldoes your model seem to fit? Experiment with different parameters until you get thebest fit you can. You may have to try a number of different models before finding onethat is satisfactory.

(d) The population of Zimbabwe is currently about 12.6 million. According to your model,about what percent of this population has AIDS? (Note that this could be significantlyless than the number that are infected with the HIV virus but have not yet developedAIDS).

(e) Suppose the percentage of the entire world population that has AIDS is the same as thepercentage in Zimbabwe. The world population in 2005 is estimated to be about 6.4billion. How many people would have AIDS under this assumption?

(f) According to the Population Reference Bureau, about one quarter of the population ofZimbabwe already has the HIV virus. How does this fact affect your model?

9. In the section on logistic models, we note that a logistic graph has an inflection point whenL(t) = 1

2M , and that this occurs when t = ln(b)r . You can see why this is by solving the

equation

12M =

M

1 + be−rt

for t. Solve this equation for t, writing out your solution step by step. You can consultExample 1 in the text to help. As an additional hint, recall that ln(bn) = n · ln(b), accordingto the properties of logarithms.

10. In the text, we estimated that currently, there is about 200,000 square feet for each personin the world. We also estimated, based on a logistic growth model, that the world’s humanpopulation will eventually level off at around 10.4 billion. If and when the population reaches10.4 billion, how many square feet will each person have? (Hint: There is an easy way and ahard way to do this!)

11. In Example 1, we considered the growth of yogurt in a 32 ounce jar, starting with 1 ounceof yogurt culture. In that example, it took 4.2 hours for the amount of yogurt to reach 31.5ounces. Suppose you decided to speed up the process by adding more yogurt at the beginningof the process.

(a) How would the function L from Example 1 change if we started with 2 ounces insteadof 1 ounce of yogurt?

(b) Using the new logistic model from part (a), how long would it take for the amount ofyogurt to reach 31.5 ounces?


Other Families of Functions: Activities and Class Exercises1. The Urbanization of Botswana. The table and graph below show the urbanization of the

population of Botswana over time. The dependent variable is the percent of the populationwhich is classified as urban.

Year 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000Percent Urban 0.4 0.8 1.8 3.9 8.1 12.8 18.5 28.6 42.4 47.2 49Percent of Urban Population - Botswana

0

10

20

30

40

50

60

1940 1950 1960 1970 1980 1990 2000 2010

(a) Create the best logistic model for this data that you can. You may use the logisticregression feature on your calculator, if you wish.

(b) For each of the three parameters in your equation from part (a), explain what theparameter means in the context of urban population in Botswana.

2. The Ozone Hole Revisited. In the Chapter Five activities, you may have considered datarelated to the size of the ozone hole over the Antarctic. In this activity, we will reconsiderthese data, using some of the other families we have discussed since then. Refer to ChapterFive for the original data and scatterplot.

(a) Recalculate the linear regression equation for this data, letting x = 2 represent theyear 1982. Also, calculate the power regression equation. Based on both the correlationcoefficient and the shape of the scatterplot, which model do you think is better? (NOTE:Remember for the power regression you must exclude any points with 0 as an x or ycoordinate. This is why we suggested you let x = 2 denote the year 1982.)

(b) Based on the scatterplot, it seems that after 1989, the data are approximately linear,but with a smaller slope than the data show up to that point. Find the linear regressionmodel for the data but use only the years starting with 1989 (x = 9). Give the equationand the correlation coefficient.

(c) Graph your data together with all three models (both linear models and the powermodel), over a range including all the data points. Which model do you think wouldgive the most accurate predictions for the next 10 to 20 years? Explain why you thinkso.

(d) Both linear and power models (with positive exponent) have y values which get largerand larger as x increases. However, the ozone hole can only get so big. Find a reasonableupper limit for the size of the ozone hole (in square kilometers). Knowing the surfacearea of the earth might be useful.

222

(e) One type of equation that can be used to model data with an “upper limit,” is atransformed exponential function. One way to find such a model is to transform thedata, and then use exponential regression. In our case, we can do this in two steps (orone step with two operations). We can subtract the upper limit from all the data points(this makes the data all negative with an upper limit of 0), and then multiply the databy −1. (On a TI-83, you could do this by creating a new list with values −(L − U)where L is the list with the original data and U is the upper limit).

i. Do the transformation of the data as described, using your estimated upper limit.ii. Find the exponential regression equation for the transformed data. Use the original

x-values with the transformed y-values. Keep track of the correlation coefficient.iii. Transform your equation to an equation that will work for the original model.iv. Graph your transformed equation together with the data. Decide which of the four

models you have created is the best model.v. Repeat the previous steps using a couple of different upper limits. Try one that

is just slightly higher than the largest area given in the actual data. How doeschanging the upper limit seem to effect the correlation coefficient and the graphicalfit?

(f) Based on what you have done so far, which of the functions that you have developedseems to be the best model? What is your best estimate as to how large the ozone holewill eventually become? Note that this estimate is based entirely on numerical data. Itwould definitely be helpful if we had some scientific information or explanation of howor why the ozone hole was growing.

3. World Hunger In 1996, the United Nation’s Food and Agriculture Organization conveneda World Food Summit in order to begin to comprehensively address the problem of worldhunger. The summit established the goal of reducing the number of chronically hungry peoplein the developing world from 819 million in 1990 to half that by 2015. The FAO’s 2002 report,The State of Food Insecurity in the World: 2002, noted that progress towards the goal hasbeen much slower than hoped for.123

Worldwide, the latest estimates indicated . . . [that there were 799 million un-dernourished people in the developing world in 2000], . . . a decrease of just20 million since 1990-92, the benchmark period used at the World Food Summit(WFS). This means that the average annual decrease since the Summit has beenonly 2.5 million, far below the level required to reach the WFS goal of halving thenumber of under nourished people by 2015. It also means that progress would nowhave to be accelerated to 24 million per year, almost 10 times the current pace, inorder to reach that goal.

The report includes the following graph.

(a) The graph includes two different projections, one a “business as usual” scenario and onereaching the 2015 target.

i. What family of functions do both scenarios seem to be using?ii. Assuming the business as usual projection uses a linear model, what is the slope of

the line, based on the information given in the quote from the report?iii. Find an equation for the business as usual linear model. You can choose which year

you would like to be represented by x = 0; any year from 1995 to 2000 might beconvenient.

iv. Using this model, in what year would we reach the goal of around 410 million hungrypeople in the developing world?

123Reprinted with permission of the Food and Agriculture Organization of the United Nations. The full report canbe accessed at www.fao.org


v. Consider the second scenario. What slope would be required for a linear model ifwe are to reach the 2015 goal by 2015, starting from a value of 799 million in 2000?

vi. Using your second model, if the trend continued indefinitely, when would hunger inthe developing world be eliminated?

(b) Find a piecewise linear function which models the entire graph, from the year 1970 on,using the “On track” scenario where the goal of 410 million is met by 2015. Again, youhave a choice of which year you want to correspond to x = 0. You can divide the graphinto 3 linear ‘pieces’ and will need a separate equation for each. You could consider theyears from 1980 to 1995 as one piece. However, you already know the slopes for twoof these pieces. It will probably be most convenient to let x = 0 represent either thestarting year (1970), or one of the years dividing the pieces.

4. Selecting a Family. Each of the situations below might be modeled by one of the familieswe have discussed so far, possibly including a function transformation. This activity willinvolve the first step in the modeling process given in Figure 1 of chapter nine, reproducedbelow. Specifically, for each situation, you should:

1. Is the situation more likely to be modeled by an empirical model, or by a theoreticalmodel. Explain why you think so.

2. Discuss which family or families would most likely make a good model for the situation.Explain why you think so.

3. For one of the families you picked in 2, discuss whether the parameters have a particularmeaning, and if so, explain what that meaning is as specifically as possible.

Feel free to be creative! Also, try to be as thorough as you can in your discussion, eventhough you will not have complete information.

(a) You are a member of the government in a developing country that is currently in the verybeginning stages of a famine caused principally by a drought. You are very concerned

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Figure 7: The Mathematical Modeling Process

about the number of people in your country at risk from starvation. As part of yourplanning for dealing with the situation, you decide to create a model which wouldestimate the total number of deaths N from the famine as a function of the time t (say,in months), assuming that your country receives little or no aid from foreign sources.

(b) You are a member of the Marketing Division of the Hewlett-Packard (HP) Corporation.The company is interested in projecting revenues from its sales of laptop computers.Part of this will involve projecting the “average real price” P (t) of laptop computers asa function of the time t in years. Laptops (or portable computers) were first marketedin the early 1980’s, and you have data on laptop prices both for HP and for several othermajor computer manufacturers going back to at least 1995.

(c) You work for the Social Security Administration. Your assignment is to find a modelwhich will predict the number of people over the age of 65 as a function of time. Youare principally interested in predictions for the years 2010 through 2050. You of coursehave complete access to all U.S. census data, and are told to assume current policieswith regards to benefits and eligibility will continue.

(d) You work for the U.S. Department of Agriculture (USDA). A new President has justbeen elected, and one of her campaign issues involved addressing the disappearance ofthe family farmer. She has asked the USDA do a study which is to include the historyof small family farms in the U.S. and projections on the number of small family farmsshould current trends continue.

5. Canadian Oil Production. Open the Fathom file Canadian Oil Production Data.You should see a case table, three sliders, and a graph which contains a scatterplot and thegraph of a Hubbert curve. The scatterplot is based on data for Canadian crude oil productionfrom 1980 through 2003. The independent variable is the years since 1980. The units for oilproduction is quadrillions of BTUs.

(a) The sliders represent the three parameters M , b, and r for the Hubbert curve model,as described in the text. These sliders are linked with the Hubbert curve shown in thegraph. Play with the values of the sliders until the Hubbert curve fits the scatterplot asclosely as possible. Record your values for the parameters M , b, and r.


(b) Recall the M represents the total area under the curve or the total amount of oil thatwill ever be produced. Given your value of M , what is the total amount of oil thatCanada will eventually produce?

(c) According to your Hubbert model, in which year will or did Canadian oil productionreach its peak?

(d) Canadian oil production was about 10 quadrillion BTU in 1980. After the peak year,production will eventually diminish to that level again sometime in the future. Estimatewhich year this will be (Hint: You can use the symmetry of the Hubbert curve aroundthe peak to help).

6. World Oil Production. Open the Fathom file World Oil Production Data. You shouldsee a case table, three sliders, and a graph which contains a scatterplot and the graph of aHubbert curve. The scatterplot is based on data for world crude oil production from 1980through 2003. The independent variable is the years since 1980. The units for oil productionis quadrillions of BTUs.

(a) The sliders represent the three parameters M , b, and r for the Hubbert curve model,as described in the text. These sliders are linked with the Hubbert curve shown in thegraph. Play with the values of the sliders until the Hubbert curve fits the scatterplot asclosely as possible. Record your values for the parameters M , b, and r.

(b) Recall the M represents the total area under the curve or the total amount of oil thatwill ever be produced. Given your value of M , what is the total amount of oil thatCanada will eventually produce?

(c) According to your Hubbert model, in which year will or did world oil production reachits peak?

(d) World oil production was about 10 quadrillion BTU in 1980. After the peak year,production will eventually diminish to that level again sometime in the future. Estimatewhich year this will be (Hint: You can use the symmetry of the Hubbert curve aroundthe peak to help).

11. Adding Things Up

The first lesson of economics is scarcity: There is never enough of anything to satisfyall those who want it. The first lesson of politics is to disregard the first lesson ofeconomics.— Thomas Sowell

Energy Consumption

In Chapter Two and again in Chapter Ten, we considered energy production, and looked atso-called Hubbert curves. We saw that oil production in the U.S. and Norway had already peakedand is on the decline, and that world production is predicted to decline starting around 2010.124

We suggested that a global energy crisis might be in our future.Of course, if there was not a huge demand for energy there would be no crisis. We need to

consider not only the history of energy production, but also the history of energy consumption andsee if we can make some reasonable predictions based on the known data. Figure 1 shows a graphof Total Annual U.S. Fossil Fuel Consumption from 1982 to 2002. The scatterplot has a stronglinear trend, and the regression line is shown. The equation is y = 0.18984x+10.5832, where x = 0represents 1980 and the correlation coefficient is r = 0.986.Total U.S. Fossil Fuel Consumption 1982 to 2002Billions of Barrels of Oil Equivalent

101112131415161980 1985 1990 1995 2000 2005

Figure 1: U.S. Energy Consumption

Total consumption includes consumption of oil, natural gas, and coal. The consumption ismeasured in oil equivalent. In other words, if we used only oil to satisfy our fossil fuel consumption,instead of a combination of oil, gas, and coal, this is how much oil we would need to get theequivalent amount of energy. According to the graph, in 2000, the U.S. used fossil fuels equivalentto about 14.7 billion barrels. How much energy is this?

124Keep in mind that this is just one prediction. There are many organizations and companies involved in predictingfuture oil production, and as you can imagine, their predictions vary widely. See for example Has Global OilProduction Peaked? at http://www.csmonitor.com/2004/0129/p14s01-wogi.html.

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11. Adding Things Up 227

A barrel of oil is 42 gallons, which gives you a rough idea of its volume. However, the amount ofenergy is really the important characteristic. Energy is usually measured in British Thermal Units,BTU for short. The BTU is defined to be the amount of energy required to raise the temperatureof 1 pound of water 1 degree Fahrenheit. To give you a more meaningful picture, eight gallons ofgasoline contain about 1 million BTU. If your car gets 25 miles per gallon, then 8 gallons wouldget you about 200 miles. This means 1 million BTU is enough to drive your car 200 miles.

Gasoline is refined from crude oil. One barrel of crude oil contains about 6.2 million BTU.125

So, a barrel of oil is enough for you to drive your car about 6.2× 200 = 1240 miles.In 1999, the average single-family residence in the United States consumed a little over 100

million BTU of energy, and the per capita consumption of energy was about 351 million BTU. Thelatter figure includes energy used for all purposes, not just residential use. Since 351/6.2 ≈ 57,U.S. citizens used, on the average, about 57 barrels of oil in 1999. This is enough to drive about70,000 miles in a car getting 25 mpg.

Now, suppose we want to use our regression equation to estimate how much total energy the U.S.will use over the next 10 or 20 or 50 years. Our regression equation tells us only the consumptiony in any one given year x. To get the total consumption, say for the years 2005 through 2015 wecould compute the yearly consumption for each of these 11 years using the equation, and then addthese up. This would be somewhat tedious, but doable in a few minutes. On the other hand, if wechanged the range of years we would have to do our calculations over. If we wanted 50 years worthof production, the calculation would be a big pain (and would be highly susceptible to error).

Is there a better way?

Fortunately, the answer is yes. Based on what we know about linear functions, we will be ableto easily develop a way of adding up the consecutive values of a linear function. Let’s illustratewith a simple example.

Suppose you want to add up the odd numbers from 1 through 101. Let’s denote this sum byS. So, we want to know the sum

S = 1 + 3 + 5 + 7 + · · ·+ 101

Notice that each number is two more than the previous number. We could consider thesenumbers as the y-values of the linear function y = 2x− 1 of slope 2, for integer x-values from 1 upto 51. Now, we’ll do a little arithmetic trick to simplify this sum. Let’s write it down two ways,frontwards and backwards, and then add vertically.

S = 1 + 3 + 5 + · · ·+ 97 + 99 + 101S = 101 + 99 + 97 + · · ·+ 5 + 3 + 12S = 102 + 102 + 102 + · · ·+ 102 + 102 + 102 = 51 · 102

Solving for S we get that

S =51 · 102

2= 2601

This was certainly easier (and less error prone) than adding up all 51 numbers by hand or evenwith a calculator.

Why did this work? The key is that we always got the same number, 102, when we addedvertically. This happened because each number differed by 2 from the numbers before it and afterit in the list. In the first line, the numbers increased two each time, and in the second line theydecreased by two each time. This, in turn, occurred because the numbers we were adding wereconsecutive y-values of a linear function with slope 2.

Notice that, in the final calculation of our sum S, all we had to know is how many numberswere being added (51 in this case), and the sum of the first and last numbers (1+101 = 102). The

125Information in this paragraph is from the Department of Energy’s Energy Information Adminsitration athttp://www.eia.doe.gov/neic/infosheets/apples.htm.

228

sum S is the product of these two numbers divided by 2. The same procedure would work for anylist of numbers that are consecutive values of a linear function. Let’s use this idea to predict thetotal U.S. energy consumption for the years 2001 through 2050.

From our regression equation y = 0.18984x + 10.5832, we have that U.S. consumption in 2001and 2050 are estimated to be

y = 0.18984(21) + 10.5832 = 14.57y = 0.18984(70) + 10.5832 = 23.87

Consumption for each year will be 0.18984 more than the year before, since the consumption valuesare given by consecutive values of the linear equation y = 0.18984x + 10.5832. We are adding 50consumption values, so the sum will be

50(14.57 + 23.87)2

=50(38.44)

2= 961

In other words, if current trends continue, the U.S. will use the equivalent of 961 billion barrelsof oil during the first half of the 21st century. How much is this really? It is estimated that therewere 1.266 trillion barrels of proven oil reserves in 2004,126 up from 0.936 trillion barrels 10 yearsbefore.127 Of course, the U.S. is using coal and natural gas as well as oil. In fact, from 1990 to2000, about 45% of U.S. fossil fuel energy consumption came from oil. This is the good news. Onthe other hand, the U.S. currently uses “only” about 40% of the world’s energy. The rest of thecountries of the world consume the remaining 60%, and many of these countries are seeing theirannual consumption increase dramatically, much more quickly than the U.S. consumption.

If we assume the ratio of oil consumption to total fossil fuel consumption for the world atlarge is similar to the ratio for the U.S., that the ratio of U.S. to world consumption remains atabout 40%, and that current trends continue, our calculations would imply that roughly 80% ofthe known world supply of oil will be gone in 50 years.

Obviously, if more oil is discovered, this would increase the time to depletion. Some estimatesclaim that actual recoverable oil reserves may be as high as 3 trillion barrels, about two andone-half times the current known reserves. Also, there are widely varying estimates for trendsin future world energy consumption. Finally, under Hubbert’s assumptions, annual productionwould decrease rapidly from the peak and then level off as production nears zero. So, our rate ofproduction (and necessarily consumption) would have to decline, instead of continuing to grow, aswe deplete the resource.

For these and other reasons, predicting the date of oil depletion is tricky. To give one otherexample, estimates from the Hubbert Peak of Oil Production website predict that yearly worldproduction will drop from the current level of about 25 billion barrels128 to below 1 billion barrelssometime between 2050 and 2075, with total depletion around the year 2100 or earlier.

Arithmetic Sequences and Series

In our discussion of energy consumption, we used a linear equation to model U.S. consumption,evaluating the function only at integer values representing years. When the domain of a functionis restricted to the set of positive integers, or some subset of the positive integers, mathematiciansoften refer to the function as a sequence.

In Chapter Four, we introduced the “f(x)” notation as the standard way to denote functionsin symbolic form. We could use the same notation to denote sequences, as sequences are simplyfunctions, but mathematicians historically have used another notation. To illustrate with an ex-ample, the function f(x) = 2x − 1 in sequence notation would be denoted an = 2n − 1. Thenotation works the same way as the function notation, except that we usually use n as the inde-pendent variable instead of x, and n appears as a subscript instead of in parentheses.129 However,

126http://www.csmonitor.com/2004/0129/p14s01-wogi.html127The data is available at a variety of locations, including http://www.geocities.com/combusem/ENERCIA.HTM.128This is based on data from http://www.eia.doe.gov/neic/infosheets/crudeproduction.htm.129Why is n used? Positive integers have also traditionally been called Natural Numbers, thus the n


evaluating values for a sequence would still be done as with functions. For example, if an = 2n−1,we have

a1 = 2 · 1− 1 = 1a2 = 2 · 2− 1 = 3a5 = 2 · 5− 1 = 9

and so on. The values of the sequence are called terms. We would call a1 the first term, a2 thesecond term, etc., and an the general term or the nth term.

Since the values for the independent variable in a sequence are usually the positive integers, asequence is sometimes represented simply as a list of terms. For example, the sequence an = 2n−1could be represented by

1, 3, 5, 7, . . .

where “ . . . ” simply means that the pattern continues. This is sometimes useful, especially whenthere is no obvious symbolic representation for the sequence. For example, for the sequence1, 1, 2, 3, 5, 8, 13, . . . we can with a little thought see that the pattern is “add two consecutiveterms to get the next term.” However, it is not at all obvious how we might write a symbolicrepresentation for the general term an.

A sequence that is a linear function, like an = 2n−1, is called an arithmetic sequence. Whatwe have called the slope is called the common difference when the context is a sequence.

Series

A series is the sum of some number of terms of a sequence. Series can be denoted as we didabove, for example

S = 1 + 3 + 5 + 7 + · · ·+ 101

We call this the expanded form of the series. The terms are written out consecutively with“ · · · ” used to avoid having to write out every term. Alternatively, we often use what is knownas the “sigma notation” to denote series. We introduced this notation briefly in Chapter Three.Using sigma notation, the series above can be written

50∑

i=1

2i− 1

The variable i is called the index, and 2i − 1 is the general term of the series. The notationsignifies that we will add up the values of the sequence given by the general term starting with thefirst term (i = 1) and going up to the fiftieth term. If we wanted to add up the terms starting withthe 4th term and going to the 21st, we would write

21∑

i=4

2i− 1

In general, we let Sn denote the sum of the first n terms of any sequence an. In other words,

Sn =n∑

i=1

an

Using this notation, the sum of an arithmetic series an would be

Sn =n(a1 + an)

2Here, n is the number of terms we are adding, and a1 +an is the sum of the first and the last terms.

230

Example 1. What is the sum of the following series.

1.∑70

i=1(3i− 1)

2. 11 + 17 + 23 + · · ·+ 119

Solution:

1. Since the terms are given by the formula 3i − 1 which is a linear function of i, this is anarithmetic series with 70 terms. The first term is a1 = 3 · 1 − 1 = 2 and the last term isa70 = 3 · 70− 1 = 209. So the sum is

Sn =n(a1 + an)

2=

70(2 + 219)2

=15470

2= 7735

2. Here, the first term is 11 and the last term is 119. We can see that the terms are increasingby 6 each time. We need to find the number of terms. One way to do this is to first find ageneral formula for an. Since the common difference is 6, the general formula for the sequenceis of the form an = 6n + k for some k. If n = 1, we get 11 = a1 = 6 · 1 + k, so that k must 5.So, an = 6n + 5. You can check this expression on the first few terms.

To find out how many terms n there are, set an = 6n + 5 = 119. Solving this for n, we getn = 19. So, the sum is

Sn =n(a1 + an)

2=

19(11 + 119)2

= 1235

Savings Plans and Investments

So far, we have looked at how to add up consecutive values of a linear function or arithmeticseries, and have introduced some general notation related to sequences and series. Next, we willlook at adding up the consecutive values of an exponential function. A sequence where the termsare given by an exponential function is called a geometric sequence and a sum of the terms of ageometric sequence is called a geometric series. We will introduce these in the context of savingsplans.

Suppose you are really planning ahead and want to start saving money for your retirement.Suppose decide to put a certain amount aside each month, say $100, into a savings account. Let’sassume that interest in the account is compounded monthly and denote the monthly interest rateby i. We know that when interest is compounded periodically, the balance in the account growsexponentially according to the formula B = P (1 + i)t, where P is the amount invested, i denotesthe periodic interest rate, and t the number of periods. In our context t will denote the number ofmonths, and i is the monthly interest rate.

Let’s assume next that you want to retire in 40 years. If you make your first payment at the endof this month, it will grow at the monthly interest rate i for 40 years, or 40 · 12 = 480 months. Atthe end of this time, it will be worth 100(1+i)480 dollars. If the annual interest rate on your accountis 6%, then i = .06/12 = .005 and your first payment will be worth 100(1.005)480 = $1095.75.

Now, this is not a whole lot of money relative to supporting you in your old age, although it isover ten times the $100 you invested. Of course, this is only one of your monthly deposits. Whatwould happen if you made deposits every month for 40 years?

Well, each deposit would grow according to the same formula, except that each deposit willbe growing for one month less than the previous deposit. Your second deposit will grow for 479months, your third for 478 months, etc.


What will the account be worth on the day you make your last deposit?

On the day you make that last deposit, it will not have earned any interest, so it adds only$100 to the value of your account. The second to last deposit earns interest for one month, so itwill contribute 100(1.005)1 to the total. Adding up all the deposits, the total in the account wouldbe

S = 100 + 100(1.005)1 + 100(1.005)2 + · · ·+ 100(1.005)478 + 100(1.005)479 + 100(1.005)480

Again, this would be a big pain to calculate. There are actually 481 terms to add up. Is therea short cut in this case, as there was when we were adding up terms of an arithmetic sequence?

Yes, in fact, there is. In this case, the aspect that will help us is that the terms we are addingup are consecutive values of an exponential function, namely the function y = 100(1.005)t. In thecontext of sequences, we will write an = 100(1.005)n, and we call such a sequence a geometricsequence. Geometric sequences are simply exponential functions where the independent variable isrestricted to the positive integers. What we previously called the growth factor for the exponentialfunction will now be called the common ratio of the geometric sequence. The reason for this isthat if you take the ratio of any two successive terms, the result is equal to the growth factor. Inour example, for any positive integer n, we have

an+1

an=

100(1.005)n+1

100(1.005)n= 1.005

Another way to express this is that every term in our sequence is 1.005 times the previous term.In symbols, a2 = 100(1.005) = 1.005a1, a3 = 100(1.005)2 = 1.005a2, etc.

In general, in a geometric series with common ratio r and first term a1, we know that an+1 =r · an and that an = a1r

n−1. Notice that the exponent is one less than the number of the term.Let’s see how we can use the “exponential nature” of geometric sequences to help us find the

sum. Rather than develop the formula only for our example, we will develop a formula for the sumof the first n terms of any geometric series an = a1r

n−1. We are looking for a formula for

Sn =n∑

k=1

a1rk−1 = a1 + a1r + a1r

2 + · · · a1rn−2 + a1r

n−1

If we multiply both sides of this equation by r, and note that r · rk = rk+1, we get

r · Sn = a1r + a1r2 + ·a1r

n−1 + a1rn

Notice that the right hand side of this equation is the same as in the previous equation exceptthat the second has an additional term a1r

n and is missing the term a1. If we write the twoequations together, lining up like terms, and subtract the second equation from the first, all theterms except these two cancel and we get the following.

Sn = a1 + a1r + a1r2 + · · ·+ a1r

n−2 + a1rn−1

r · Sn = a1r + a1r2 + · · ·+ a1r

n−2 + a1rn−1 + a1r

n

Sn − rSn = a1 − a1rn

Now, we can factor each side and solve this equation for Sn.

Sn(1− r) = a1(1− rn)

Sn =a1(1− rn)

1− r(1)

This gives us our formula for the sum of a geometric series.

232

Your Nest Egg as a Geometric SeriesLet’s apply this to our retirement plan. We want to know the sum

S = 100 + 100(1.005)1 + 100(1.005)2 + · · ·+ 100(1.005)478 + 100(1.005)479 + 100(1.005)480

Our first term is a1 = 100, r = 1.005, and n = 481. Substituting these values into (1), we have

S =100(1− 1.005481)

1− 1.005=

100(1− 11.01224)−0.005

= $200, 244.80

So, you would have saved $200,244.80 for your retirement. Is this enough?As a quick estimate, suppose your account continues to pay 6% yearly interest, compounded

monthly, and you want to live off the interest without taking any of the $200,244.80 principalor balance. If we take 0.005 times $200,244.80, we would get a monthly income of $1001.22 or$12,014.69 per year. This is not a huge amount to live on, especially considering that a dollar inforty years will be worth a lot less than it is today due to inflation. You might need to considerhow much social security you will be entitled to, if there will be funds available to you from othersources (e.g. inheritance), and what you expect your expenses will be. For instance, perhaps youwill have long since paid for a house, and so will have no rent or mortgage expense. A morecomprehensive evaluation of your retirement plan would need to include these and other factors.

Owning a home is the biggest financial commitment most people make during their lives, andsince it bears on a person’s long term financial outlook in a number of ways, let’s consider thatnext.

Owning Your Own Home

Sooner or later, you will probably want to leave the ranks of renters (if you haven’t already)and become a home owner. There are many financial advantages to home ownership, besidesthe emotional and psychological benefits that most people experience. For one, a home typicallyappreciates in value over time so that, even if you only live in the home for a few years, you canprobably sell it for more than you paid for it. It is in this sense that buying a home is consideredan investment. Secondly, most people do not have enough cash on hand when they want to buytheir first house, and so they must borrow the money by taking out a mortgage. While having toborrow money might seem like a bad thing to have to do, the good news is that all the interestyou pay on the mortgage is tax deductible. In other words, you do not pay taxes on any fundsthat you use to pay mortgage interest. As we will see, this can save you a considerable amount ofmoney. Finally, depending on which region of the country you live in, buying a home can result ina smaller monthly payment than renting a similar residence.

The first step in buying a house is to find out how much you can afford to borrow, given yourhousehold income and the funds you have available for a down payment. Mortgage lenders typicallywill not write mortgages where the monthly payment is more than 40% of monthly income, andusually recommend a ratio of no more than 30%. The monthly payment should include real estatetaxes and home insurance, in addition to the payment on the loan. Knowing the amount you canafford for a monthly payment, the duration of the loan, and the available interest rate determineshow much you can afford to borrow. Most home mortgages are paid off over 30 years or 360months, but some may be for shorter periods, 15 years being common. We will assume paymentsare made monthly, which is the case in the vast majority of mortgages. Let’s look at how thisworks mathematically.

We will develop the general formula relating the monthly payment, loan amount, loan duration,and interest rate. The basic idea is that, each month, you must pay interest on the remainingbalance of the loan, using the interest rate i. You also want some of your monthly payment to gotowards paying off your debt, so that you have less remaining to be paid off than the month before.After 360 months, or whatever the loan duration is, all the balance is to be paid off.

The details are a little messy, but are really just based on common sense, the basic idea ofinterest, and what we already know about geometric series. We will use the following notation forthe variables involved so that we can develop our formula.


L = Loan amountM = Monthly payment

i = monthly Interest rateN = Number of months duration

Also, we will let B(k) stand for the balance of the loan remaining to be paid off after k monthlypayments have been made. B(0) = L, the amount remaining to be paid before any payments aremade. B(N) = 0, since the whole loan is to be paid off after N months.

Now, at the end of the first month, you will pay the bank M dollars, since M is your monthlypayment. Of this, iL dollars goes to pay interest, and the remaining part of your payment, M− iL,goes to pay off the principle. So, after this first payment, the remaining balance is

B(1) = L− (M − iL) = L + iL−M = L(1 + i)−M

Now, for the second month, you will pay interest on the current balance B(1). So, out of yoursecond monthly payment, the amount going for interest is i ·B(1) = i(L(1 + i)−M). The amountgoing to pay off the loan is thus M − i · B(1). Subtracting this from B(1) gives the remainingbalance after 2 payments, in other words B(2). We have

B(2) = B(1)− (M − i ·B(1)) = B(1) + i ·B(1)−M

= B(1)(1 + i)−M

= (L(1 + i)−M)(1 + i)−M

= L(1 + i)2 −M(1 + i)−M

Let’s do one more month. For the third payment we have

Amount of interest paid = i ·B(2)Amount of principal paid = M − i ·B(2)

Remaining balance = B(3) = B(2)− (M − i ·B(2))= B(2) + i ·B(2)−M

= B(2)(1 + i)−M

= [L(1 + i)2 −M(1 + i)−M ](1 + i)−M

= L(1 + i)3 −M(1 + i)2 −M(1 + i)−M

= L(1 + i)3 −M((1 + i)2 + (1 + i) + 1)

Now, it looks like we have a pattern that we can use to find B(k) for any k.130 After k payments,the remaining balance would be

B(k) = L(1 + i)k −M [(1 + i)k−1 + (1 + i)k−2 + · · ·+ (1 + i) + 1]

Since we want the loan to be paid off after N months, we would have B(N) = 0. This wouldbe true if and only if

L(1 + i)N = M((1 + i)N−1 + (1 + i)N−2 + · · ·+ (1 + i) + 1) (2)

130We are assuming that this pattern will continue for all possible k. Although you might be convinced that ourformula is OK, we should note that we have not proven mathematically that our formula works. For now, we willsimply note that this can be done, although we will not go into the details here

234

Finally, we will use what we know about geometric series to simplify. Ignoring the M , the righthand side of the equation above is a geometric series, written “backwards,” with common ratior = (1 + i), first term 1, and a total of N terms. Using our formula (1) for the sum of a geometricseries, we have that

(1 + i)N−1 + (1 + i)N−2 + · · ·+ (1 + i) + 1 = 1(1−(1+i)N )1−(1+i)

= 1−(1+i)N

−i

= (1+i)N−1i

Substituting this into equation (2) above, we get

L(1 + i)N = M · (1 + i)N − 1i

Since, in our case, we will know the monthly payment M we can afford, we will solve this forL. We have

L = M · (1 + i)N − 1i · (1 + i)N

= M · 1− (1 + i)−N

i(3)

In other cases, we may wish to solve the equation for M in order to find the monthly paymentfor a given loan amount. We get

M = L · i

1− (1 + i)−N(4)

This was a little bit of work, but now we have a formula which we can use to find out howmuch we can borrow knowing the interest rate, the amount we can afford to pay monthly M ,and the duration of the mortgage N . Let’s say we are making $45,000 per year and that wewant to use at most 30% of our income to make our monthly payments. Our monthly income is$45, 000/12 = $3750 and 30% of this is $1125. Let’s suppose real estate taxes for the types ofhouses we might consider are around $100 per month and that insurance will cost about $25 permonth. This leaves us $1000 per month to pay off the loan. This will be our M .

Interest rates have fluctuated substantially over time, from recent lows of around 6% to over12%. As an example, let’s assume an annual interest rate of 9%, which translates to a monthly rateof i = 0.0075. If we plan on a standard 30 year mortgage, the duration will be N = 360 months.The loan amount we can finance would thus be

L = 1000 · 1− (1.0075)−360

0.0075≈ $124, 282

We can afford a house of about $125,000.

We can also use the formula for the geometric series (1 + i)N−1 + (1 + i)N−2 + · · ·+ (1 + i) + 1to simplify our expression for B(k). We have

B(k) = L(1 + i)k −M [(1 + i)k−1 + (1 + i)k−2 + · · ·+ (1 + i) + 1]

= L(1 + i)k −M · (1 + i)k − 1i

(5)

Before we go on to consider the investment aspects of owning a home, it is worth considering,and is surprising to most people, the interest paid for such a mortgage. In our example, you wouldbe paying $1000 per month every month for 30 years to pay off the mortgage. This means you will


pay a total of $360,000 for your house, which cost $124,282. This is almost three times the priceof the house!! In interest, you will pay 360, 000− 124, 282 = 235, 718 dollars. How is this possible?Perhaps more importantly, how can this possibly be a good thing to do?

To give a partial answer to the first question, note that, out of your first $1000 payment, youare going to pay 0.0075 ·124, 282 = $932.12 in interest, leaving only $67.88 to go towards paying offyour debt. This may seem surprising, but it is in general true that on a 30 year mortgage, only avery small fraction of your first payment (in fact, all of the payments for many years) goes towardspaying off the principal of the loan. You can use the formula for B(k) given above to experimentand find out how slowly the balance goes down during the initial years of the mortgage.

On the other hand, recall that you do not have to pay taxes on the interest you pay, whereas ifyou were renting, you have to pay taxes on all the money you would use to pay for housing. Evenif you are in the lowest current tax bracket, this means you will save 15% of the amount you spendon interest. In our example, this means you save (0.15) · $235, 718 = $35, 357.70 in taxes. If youare in the 28% tax bracket, you would save almost twice as much.

Also, keep in mind that you would pay for housing, whether you rent or buy. If you wererenting, you might pay a total of $360,000 or even more over 30 years, and not get any tax break,and you would own nothing as a result of your expense. After paying off your mortgage, you nowown “free and clear” a house that was worth $125,000 when you bought it, and may well be worthseveral times that amount 30 years later.

Housing as an investment

How much can you expect your house to grow in value over time?

In the United States, the Office of Federal Housing Enterprise Oversight (OFHEO) publishesa Housing Price Index (HPI) which tracks how the value of a typical single-family home changesover time in various regions of the country, as well as state by state. You could use the HPI forthe region where you live to get at least some idea of how your house might appreciate in value,whether you plan to live there for 30 years, or only a few years.Housing Price Index for South Dakota

0501001502002501975 1980 1985 1990 1995 2000 2005

Figure 2: Housing Price Index for South Dakota

Figure 2 shows the HPI for South Dakota along with an exponential regression of the data.Although there was quite a bit of volatility in the HPI during the early 1980’s, the exponentialcurve seems to fit the data very well. The equation for the exponential regression curve is y =

236

68.265(1.04287x), which indicates that housing prices in South Dakota have increased at about4.287% per year.

Suppose you buy your house in South Dakota for $125,000. Assuming the same annual averagegrowth rate of 4.287%, in 30 years, your house would be worth $125, 000(1.0428730) ≈ $440, 370.If you sold your house at this time, you would have a considerable sum of money which you couldeither use to buy another house, or put into savings for your retirement. If you had already raiseda family, you may want to live in a smaller, less expensive home. In this case, perhaps you couldbuy a home using a portion of the $440,000 and use the remainder as savings. Alternatively, youmay decide to go back to renting.

In any case, as a result of your “investment” of $1000 per month for 30 years, you would haveclose to half a million dollars and you would have had a place to live for that period of time. Now,you might think you could have done better by just investing $1000 every month in some sortof savings plan, and this is probably true, but you would still have to pay for a place to live. Ifyou’d like to consider another scenario for comparison, you could calculate what would happen ifyou scrimped on your living conditions and rented a residence for something less than $1000 permonth for the 30 years, and then saved the extra money in a savings plan. One of the activitieswill outline this option.

Conclusion

In this chapter, we have developed easy ways to add up values of arithmetic and geometricsequences. These correspond to linear and exponential functions respectively. Since linear andexponential functions are so important and so common, this allows us to “add things up” in lotsof situations that we might run across in the real-world.

However, it is worth remembering that there are lots of (important) functions that are notlinear or exponential. Could we find an easy way, for example, to add up the consecutive values ofa power function like f(x) = x2? What about logarithmic functions?

In general, it is unfortunately much more difficult to find sums when the sequence is neitherarithmetic nor geometric. Our last resort, of course, is to just add up the specific series we have“by hand.” Fortunately, spreadsheets and even graphing calculators can make this tedious taskmuch more feasible, and there will be cases where we will have to resort to these alternate methods.The important thing is to be able to recognize when you have an arithmetic or geometric seriesand when you do not, so that you know which techniques will work, and which will not.

Reading Questions

1. What is a sequence? How is a sequence the same or different than a function, as functionsare described in Chapter Two?

2. Describe how you could tell whether or not a sequence is an arithmetic sequence.

3. Describe how you could tell whether or not a sequence is a geometric sequence.

4. For each sequence given below, decide whether the sequence is arithmetic, geometric, orneither.

(a) an = 0.6n + 1.3(b) an = 0.2n1.5

(c) an = 0.75n

(d) an = (−1)n(1.5n)(e) 8, 6.4, 5.12, 4.096, · · ·(f) 3, 7, 11, 15, · · ·(g) a1 = 1000 and each term is 1.0075 times the previous term.(h) a1 = a2 = 1, and for n > 2, each term an is the sum of the previous two terms.(i) a1 = 2.5 and each term is 0.15 more than the previous term.

5. Write out the following sums in expanded form.

(a)∑5

i=1(1.5i + 1)

(b)∑6

k=3(1.5k + 3)

(c)∑7

n=0(2n)

(d)∑8

m=4 3(1/2)n

6. Find the following sums. Note that some of the series may be arithmetic, some geometric,and some neither.

(a) 1 + 4 + 7 + 10 + · · ·+ 601(b) 1 + 4 + 9 + 16 + · · ·+ 100

(c)∑120

i=1 50(1.085i)(d) 1 + 1/2 + 1/4 + 1/8 + · · ·+ 1/212

(e)∑32

k=1(1.5k + 1)

(f)∑45

k=7(1.5k + 1)

7. In Chapter Five, one of the applications concerned annual carbon emissions from fossil energyconsumption by the U.S. commercial sector as a function of time. The linear regression linefound there was y = 21.9x + 757, where x was the number of years since 1990 and y wascarbon emissions in millions of metric tons for that year.

(a) Explain why using the sum of an arithmetic series would be an appropriate way toestimate the total amount of carbon emissions over a period of several years.

(b) Assuming the linear trend continues, estimate the total amount of carbon emissionsfrom 1990 through 2020.

8. Is the sequence 8, 8.8, 9.68, 10.648, . . . arithmetic, geometric, or neither.

237

238

9. One of the following three sequences is arithmetic, one is geometric, and one is neither. Saywhich is which.

(a) 8, 12, 30, 45, . . .(b) 4, 8, 14, 22, . . .

(c) 24, 18, 12, 6, . . .

10. Calculate the loan amount L in each of the following circumstances.

(a) The monthly payment M = $750, the annual interest rate is 6%, and the term of theloan is 30 years.

(b) The monthly payment M = $750, the annual interest rate is 12%, and the term of theloan is 30 years.

(c) The monthly payment M = $750, the annual interest rate is 6%, and the term of theloan is 15 years.

(d) The monthly payment M = $750, the annual interest rate is 12%, and the term of theloan is 15 years.

11. Based on your calculations in the previous question, is it true or not true that doubling theinterest rate on a mortgage cuts the amount you can borrow in half?

12. Suppose you want to buy a new pick-up truck that costs $30,000. The financing companyoffers you a 5-year loan at an annual interest rate of 7.78%.131 You have $4,000 cash on handto put towards the truck and want to borrow the remaining cost. What would your monthlypayment be?

13. Suppose you want to buy a nice home on the local lake for $280,000. You plan to have$60,000 cash on hand from the sale of your current home, and need to borrow the rest.Annual mortgage interest rates are at 6.2% for a 30-year mortgage and 5.5% for a 15-yearmortgage.

(a) Calculate your monthly payment for each type of mortgage.(b) In each case, calculate the total amount you would spend in interest over the duration

of the loan.(c) In each case, calculate the balance remaining to be paid off after 10 years (120 payments)

using equation (4) in the text. Do these results surprise you?(d) Assuming L, M , and i as above, graph y = B(k) for both the 15 and the 30 year

mortgages. Draw a sketch of your graphs on the same axes.(e) After 15 years, the 15-year mortgage will be paid off. What is the remaining balance on

the 30-year mortgage at this time?

14. Suppose you want to start saving for your retirement.

(a) How much would your total balance be if you saved $100 per month for 40 years in anaccount that paid 9% annual interest?

(b) How much would you have to save each month, at a 9% annual interest rate, in orderto amass 1 million dollars after 40 years?

(c) How much would you have to save each month, at a 6% annual interest rate, in orderto have 1 million dollars after 40 years?

15. What is a BTU?

16. If you drive a mini-van that gets 20 miles per gallon, how many BTU’s do you need to drive200 miles?

131Rates quoted at http://www.bankrate.com/brm/static/rate-roundup.asp as of August 18th, 2007


Adding Things Up: Activities and Class Exercises

1. Retirement Savings. In our retirement planning example in the text, we calculated thatsaving $100 per month at a bank or other financial institution for 40 years at a fixed 6%interest rate would result in an accumulated balance of $200,244.80. If you lived off theinterest, at the same 6%, you would have $12,014.69 of income per year ($1001.22 per month),and would never use the $200,244.80 principal.On the other hand, you are not going to live forever, and if you don’t care about leaving anyinheritance behind, you could use the interest and perhaps some of the principal each monthin such a way as to have nothing left after, say, 30 years (by which time you estimate thereis a high probability of being dead). How could you do this?You can think of this situation as a mortgage, where you are the one loaning the bank$200,244.80, and the bank will be paying you back over 30 years. After 30 years of monthlypayments, the bank will have paid you off; in other words, your principal would be gone. Themonthly payment will be what the bank pays you each month for your monthly income.

(a) Calculate your monthly income under this scenario. How much more do you get permonth than in the previous case where you did not touch the principal.

(b) What if you “plan” on living longer, and want the income to be paid out over 40 years?Calculate your monthly income in this case.

(c) For a long term investment plan, like the 40 year savings plan in our example, you maydecide to invest in mutual funds, which do not have a fixed rate of return, but whichhave historically grown at higher average rates, typically around 10%, rather than the6% we originally used. Calculate the accumulated amount of saving $100 per month for40 years at a 10% annual rate.

(d) When you retire (or even before) you may want to reduce the risk level of your in-vestments and so may decide to put your accumulated balance in an account that hasa guaranteed, albeit lower, fixed interest rate. Calculate how much income your new,larger balance would generate at 6% interest assuming

i. You never touch the principalii. You have the principal paid out over 30 yearsiii. You have the principal paid out over 40 years

2. Energy Production. Figure 1 of Chapter Two showed a graph of U.S. Oil Production.Production peaked in 1970 and has been generally falling since. The graph below showsproduction from 1985 to 2002. The dependent variable here is given in billions of barrels peryear.For this activity, we will assume that production declines exponentially. The table belowgives the data for the years 1985 to 2002.

Year 1985 1986 1987 1988 1989 1990 1991 1992 1993Production 3.27 3.17 3.05 2.97 2.78 2.68 2.71 2.62 2.50

Year 1994 1995 1996 1997 1998 1999 2000 2001 2002Production 2.43 2.39 2.36 2.35 2.28 2.15 2.13 2.12 2.12

(a) Find the exponential regression for this data. How well does the exponential model seemto fit the data?

(b) What does your model predict for the year 2000? Also, what does your model predictfor the year 2050?

(c) By adding the yearly productions, we can find the total production over any number ofyears. What type of series would we have if we were summing values of our exponentialmodel?

240 Total U.S. Oil ProductionBillions of Barrels per Year012341985 1990 1995 2000 2005

(d) Estimate the total U.S. oil production for 2000 through 2050.(e) Geometric series which have common ratio between −1 and 1 have the property that

you can add an infinite number of terms and still get only a finite sum. In our context,if U.S. production continued forever, there would still only be a finite amount of oilproduced. To see why this is, recall that the sum of geometric series is given by

Sn =a1(1− rn)

1− r

i. If r is between 0 and 1 and n is a large exponent (say, 100), what happens to rn?Try this for the r in your exponential model.

ii. If rn ≈ 0, then the sum is approximately S = a11−r . We call this the sum of the

infinite geometric series. Estimate the total U.S. production from now on by findingthe infinite sum of the geometric series, starting in the year 2000. How much biggeris this than the total production from 2000 to 2050?

3. Housing versus Saving as an Investment. In the text, we calculated that at a 9% annualrate of interest, we could buy a house worth about $125,000 with a monthly payment (notincluding taxes and insurance) of $1000. We also estimated that this house, located in SouthDakota, would be worth about $440,000 in 30 years. In this activity, you will investigatewhat will happen if you decide to spend less on the house and invest the difference. Let’ssuppose you decide to buy housing at $800 per month and invest $200 per month in a savingsplan.

(a) What 30-year loan amount can you afford at a monthly payment of $800 assuming a 6%annual interest rate?

(b) Assuming you buy a house for the loan amount you found in (a), and assuming the same4.287% percent growth rate in the value of your house as in the text, how much willyour house be worth in 30 years? (Hint: You should not have to recalculate 1.0428730

to do this).(c) Now, if you invest $200 per month at 9% compounded monthly for 30 years, how much

will the total balance be at the end of the 30 years?(d) Combining parts (b) and (c), what are your total assets between the house and the

savings plan after the 30 years?(e) Now suppose you had spent the entire $1000 per month on housing. Calculate the value

of the house you could buy at 6% interest and how much this house would be worth


after 30 years at the same 4.287% growth rate. How does this compare with your totalin part d).

(f) Can you think of any other information or assumptions that we haven’t included herethat might be included to make our comparison more realistic?

4. World Energy Consumption. The graph below shows estimated annual total worldenergy consumption in quadrillions of BTU.World Total Primary Energy Consumption in Quadrillions of BTU

2002503003504004501980 1985 1990 1995 2000 2005

(a) A linear regression line is included with the scatterplot. From the graph, estimate theequation of this line. You could use x = 0 to represent the year 1980.

(b) Estimate the total amount of energy consumed for all the years 1980 through 2000.(Hint:This will involve computing the sum of an arithmetic series)

(c) According to figures from the Department of Energy, total world known reserves ofcrude oil, natural gas, and coal as of 2002 would give an energy equivalent of roughly40,000 quadrillion BTU.132 Assuming continued linear growth in annual consumptionaccording to your equation from part (a), how long will it be before the world consumesa total of an additional 40,000 quadrillion BTU’s?. What year will this be?

i. Since we are concerned with the total consumption over a number of years, and notjust a single year’s consumption, we will want to use the formula for the sum of anarithmetic series

Sn =n(a1 + an)

2However, in this case, we are given the sum or total which is Sn = 40, 000. What isour unknown in this situation?

ii. We can consider a1 to be the annual production in 2002. Use your linear regressionequation to calculate this value.

iii. We do not know an since we do not know n, but we can use our linear regressionequation to express an in terms of n. Do this, and substitute the expression intothe sum formula for an. Then, solve the equation for n (Hint: the equation is aquadratic equation, so you may need to do some algebra and then use the quadraticformula). What year corresponds to the solution you found for n.

iv. In the year from the previous question, what will be the annual consumption, ac-cording to your linear model?

132Accessed from www.eia.doe.gov. However, these estimates are compiled from a number of sources, including Oil& Gas Journal, and are very tentative in nature, even though they are referred to as proved reserves

242

v. Now, suppose that by the time we reach the year from part (i), we have discoveredanother 40,000 quadrillion BTU’s of recoverable energy. If we assume the samelinear growth in consumption, how long before this additional 40,000 BTU’s is usedup?

(d) Assuming your projections are reasonable, what recommendations would you make withregards to future world energy policy?

12. Probability

When you have eliminated the impossible, what ever remains, however improbable, mustbe the truth.—Sir Arthur Conan Doyle, in The Sign of Four

I will never believe that God plays dice with the universe.—Albert Einstein

Probability, and the related field of statistics, have become increasingly important and relevantin modern society. Here are a few examples of situations where the idea of probability plays a role.

1. Medical Testing: You have tested positive for a dangerous, though not too common disease(HIV for example). How concerned should you be? (In other words, what are the chancesyou actually have the disease?)

2. DNA Testing: The term “DNA fingerprinting” first appeared in the public lexicon around1985, and has been attributed to British geneticist Alec Jeffreys. Since that time, the tech-nique has been used in the U.S. more than 50,000 times. Typically, one might read statementslike “the probability that a randomly selected person would have a DNA profile matching thedefendant’s is one in a million” or some such very small probability. The implication oftenmade is that the defendant is almost surely guilty based on the DNA evidence alone. Is thisreally a valid conclusion? In other words, if the lab says there is a DNA match between thedefendant and the sample provided by the police, how likely is it that the defendant is reallyguilty?

3. Lotteries: As of 2001, 37 states had some form of government sanctioned lottery, and about$34 billion was spent by Americans on tickets that year.133 What are your chances of winningthe jackpot in a particular lottery? How much do lottery players win in prizes, on the average,for each dollar that is spent? How much revenue do states generate from lotteries?

In this chapter, we will consider the basic concepts of probability, as well as how probabilitycan help us understand a variety of real-world situations, including those mentioned above.

The most basic examples of probability involve a random trial or experiment, a set ofequally likely outcomes, and a particular event consisting of some subset of these outcomes. Ifthe experiment is rolling one (unbiased!) die, there are six equally likely outcomes which we canlist {1, 2, 3, 4, 5, 6}. Mathematicians would call this set of all possible outcomes the sample spaceof the experiment. Now, if the event is described as “rolling a prime number,” then the outcomesrelated to this event are {2, 3, 5} and the probability of this event is 3/6 = 1/2. If the event is“rolling a 4,” then the probability is 1/6.

The “equally likely outcomes” condition is very important. For example, if you are consideringwhat tomorrow might bring with respect to precipitation, the list of outcomes might be

{rain, snow, sleet,hail, freezing rain, none},but these are not equally likely outcomes. It would certainly not be correct to say, on the basis ofthis list, that the chance of rain for tomorrow is 1

6 or that the chance of some form of precipitationis 5

6 .In the examples so far, we have given the probabilities as fractions, and this is a very common

way to represent probabilities. We can also represent probabilities as either percentages orrelative frequencies.

As a percentage, the probability of rolling a prime number when rolling a single fair die is 50%or .50. As a relative frequency, we could have said that we expect to roll a prime number 3 out ofevery 6 times.

133Data from The Statistical Abstract of the United States: 2001.

243

244 Algebra: Modeling Our World

It is important to remember that probability can never tell you for sure what will happen inany given situation, unless you know for sure that the probability is 100%. So, when we say “weexpect to roll a prime number 3 out of every 6 times,” we can really only expect this to happen insome “on the average” sense. In our next six rolls, we could get anywhere from 0 up to 6 primenumber rolls.

Three Types of Probabilities

We will classify a probability as one of three different types, depending on what information isused to calculate its numerical value. These types are empirical, theoretical, and subjective.

An empirical probability is one that is calculated on the basis of collected data. For example,suppose that medical researchers tested a new pain relief medicine on 2000 individuals and foundthat 1827 experienced at least some pain relief. We have 1827

2000 = .9135. The researchers mightreport that the probability that the drug will provide pain relief is over 91%.

Our dice probabilities are examples of theoretical probabilities. Here, we knew (or assumed)we had a fair die so that all outcomes were equally likely and simply counted the total number ofoutcomes and the total number of outcomes in the event. We did not conduct any empirical testsor collect any data. If each item in the sample space has an equally likely chance of occurring, thetheoretical probability of an event is defined as

the number of outcomes in an eventthe total number of outcomes in the sample space

.

Note that the numerator of this fraction must be at least 0, but is no larger than the denominatorsince the outcomes in any event must also be contained in the sample space. So, the theoreticalprobability of an event will always be greater than or equal to zero and less than or equal to one.A probability of zero means that the event will never happen and a probability of one means thatit will always happen.

If you did not know the die was fair, you could calculate probabilities for this particular dieempirically by rolling it a whole bunch of times and counting how many times it came up in eachof the six outcomes. If you rolled it a 200 times, you might put your results in a table like that inTable 1. For example, based on these data, we would say the empirical probability of rolling a 4with this die is 20.5%.

Outcome 1 2 3 4 5 6Frequency 25 22 6 41 32 74Probability .125 .11 .03 .205 .16 .37

Table 1: Empirical probabilities for an evidently weighted die

When a list of outcomes together with their probabilities are given in table form, as in Table1, we say we have a probability distribution. In any such distribution, all possible outcomesshould be listed and the probabilities must add up to one.

Empirical and theoretical probabilities are both legitimate ways to calculate probabilities. How-ever, you often run across examples where people are reporting probability numbers simply basedon their best judgement. For example, a political pundit might say that the probability of theRepublican presidential candidate winning the next election is better than 60%. A sports writermight say “the Lakers are a slam dunk to win the NBA title,” implying that the probability thatthey will win the championship is 100% or nearly so. These are examples of subjective probabilities.

Subjective probabilities are generally not reliable. How reliable depends more on the knowledgeand objectivity of the person involved than anything else. You might have more confidence in the60% number given in the preceding paragraph if the person reporting this number had a greatdeal of experience in presidential politics (and was not a Republican), then if he or she was aconservative talk radio host.

12. Probability 245

Counting and Listing All Outcomes

Odometer MethodFor theoretical probabilities, if we can easily count all the possible outcomes and the number

of outcomes in the event of interest, then everything is great. On the other hand, this is often notthe case. For example, if a lottery involves drawing 5 different numbers from 1 to 45, we cannotpossibly hope to count the number of possible combinations “by hand.” There are just too many.We need an easier way to count all the possible outcomes, without having to actually list them all.

Let’s illustrate with an example. Suppose a student is going to a fast food restaurant for lunch.The student likes cheeseburgers, chicken sandwiches, and fish sandwiches. He likes french friesand onion rings. He likes coke, cherry coke, sierra mist and root beer. How many different lunchcombinations are acceptable to this student?

A lunch combination has one sandwich, either french fries or onion rings, and one drink. We willfirst “list” all the possible food combinations and then we will “count” all the possible combinations.We will list all the possible food combinations by using the so-called the “odometer method.” Thismethod is analogous to how a car odometer keeps track or “counts” miles traveled. For simplicity,let’s consider an odometer with 3 positions where each position is a number between 0 and 9 (10possible numbers).

Leftmost (hundreds) position Middle (tens) position Rightmost (units) position0,1,2,3,4,5,6,7,8,9 0,1,2,3,4,5,6,7,8,9 0,1,2,3,4,5,6,7,8,9

choose one choose one choose one

The initial reading of the odometer is 000. When the car has traveled 1 mile, the reading is 001.When the car has traveled 9 miles, the reading is 009. When the car reaches 10 miles, the odometerreads 010. As the car traveled the first 10 miles, the rightmost position on the odometer went from0 to 9 then back to 0. As the care continues, the rightmost position will continue to cycle throughthese 10 digits.

Similarly, the middle position on the odometer will cycle through the digits 0 to 9 repeatedly,changing once every 10 miles or once for each complete cycle of the rightmost position. The samepattern occurs with the leftmost position. It will cycle through all 10 digits, 0 to 9, changing digitsfor each complete cycle of the middle position. In this way, the odometer would count through atotal of 1000 miles, starting at 000 and ending at 999. After mile 999, it would presumabley cycleback to 000.

If we were to try and list the odometer reading after each mile of travel we could generate alist that would look like this:

000 001 002 003 004 005 006 007 008 009010 011 012 013 014 015 016 017 018 019020 021 022 etc..........etc570 571 572 573 etc.etc.990 991 992 ...............etc........... 999

Note that if we look at the three numbers showing on the odometer, no “reading” is repeated in ourlist. For example, the reading 572 appears only once in our list. Thus, each of the 1000 differentreadings occurs once and only once. By listing the numbers in this organized fashion, we can seethere are 1000 different numbers. We might also realize this total by noting there are 10 differentoptions for each of the three positions, so there are 10× 10× 10 = 1000 total odometer readings.

We can apply this same idea to our food combinations. Just as our car odometer has 3 “slots”or positions that are to be filled with numbers, we can think of food combinations as having threeslots, namely the sandwich slot, the side order slot and the drink slot. For example, one “foododometer” reading would be as follows.

fish french fries sierra mist(sandwich) (side order) (drink)


We could say that the “reading” (fish, french fries, sierra mist) means the student ordered thiscombination for his lunch. Since there are 3 options for the sandwich slot, two for the side orderslot, and 4 for the drink slot, our food odometer will have a total of 3×2×4 = 24 possible readings,and this will correspond to 24 possible food combinations. We can list these in an organized fashionas in Table 2.

Table 2: The students twenty-four lunch possibilities

(cheeseburger, french fries, coke) (cheeseburger, french fries, cherry coke)(cheeseburger, french fries, sierra mist) (cheeseburger, french fries, root beer)

(cheeseburger, onion rings, coke) (cheeseburger, onion rings, cherry coke)(cheeseburger, onion rings, sierra mist) (cheeseburger, onion rings, root beer)

(chicken, french fries, coke) (chicken, french fries, cherry coke)(chicken, french fries, sierra mist) (chicken, french fries, root beer)

(chicken, onion rings, coke) (chicken, onion rings, cherry coke)(chicken, onion rings, sierra mist) (chicken, onion rings, root beer)

(fish, french fries, coke) (fish, french fries, cherry coke)(fish, french fries, sierra mist) (fish, french fries, root beer)

(fish, onion rings, coke) (fish, onion rings, cherry coke)(fish, onion rings, sierra mist) (fish, onion rings, root beer)

In practice, we forego writing out the list. We simply use the logic of the odometer method tocount how many possibilities there would be in a given situation without listing them. This methodwill work as long as we can identify our “slots,” can count how many options there are for eachslot, and can fill in each slot independently of how the other slots are filled in. Independentlymeans, for example, that we can order any drink with chicken and fries, or any side order withcheeseburger and cherry coke, etc. There are no restrictions like “the student will not order onionrings with a fish sandwich.” Independence implies that every possible combination is a valid one,and would appear in the list if we created the list. The odometer method of counting is also knowas the general counting method or the multiplication principle.

To summarize, any time you have several independent choices to make (slots), the number ofways to make all of these choices is the product of the number of ways of making each individualchoice. The choices are independent if how you make one choice does not effect your options in theother choices. In our example above, the choices are independent since you can choose and sideitem with any sandwich with any drink. If the restaurant had said you can only have fries if youdo not choose the fish sandwich, then these choices would not be independent, and you might bestuck having to try and list out all the options by hand.

More on Theoretical Probabilities

We can apply the multiplication principle and other counting techniques whenever we arecalculating theoretical probabilities. Recall that if each item in the sample space has an equallylikely chance of occurring, the (theoretical) probability of an event is defined as

the number of outcomes in an eventthe total number of outcomes in the sample space

.

If our sample space is all the possible lunch combinations listed above, and the event is “eatinga cheeseburger,” we need to count the outcomes containing a cheeseburger and divide that by 24(the total number of outcomes for our lunch combinations). Referring back to Table 2, we see thatthere are 8 combinations containing a cheeseburger. So the probability of eating a cheeseburger is8 divided by 24 or 1/3.

12. Probability 247

Often, we will use function notation as a shorthand for denoting probabilities. In this context,we let the inputs E be events, and use P for P robability to denote the function. So P (E) denotesthe probability that E will occur. This will be a function, since given any sample space S and anyevent E in this sample space, there is only one fraction which will give the probability P (E).

Dice Probabilities

Many classic probability problems deal with dice or cards. In fact, the foundations of probabilitywere the result of determining how to fairly divide the winnings in a game of chance. Suppose youwere determining probabilities for rolling a pair of dice. To answer this question, we need to startby finding the number of equally likely outcomes for the sum. You might guess that, since youcould have a sum of 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12 showing on the two dice, there are elevenoutcomes. However, these outcomes are not equally likely. There is only one way to get a sumof two (rolling two ones) while there are many ways to get a sum of seven (rolling a three and afour, two and a five, etc.). To predict what will happen if the process is repeated many times, theprocess must be broken down into equally likely outcomes. For rolling two dice, one gray and onewhite for example, there are 36 equally likely outcomes. We could calculate this using our generalcounting principle by noting that there are six equally likely outcomes for each die, so the totalnumber of outcomes when rolling both dice is 6 · 6 = 36. This is not a huge number, and we havelisted all 36 outcomes in Figure 1.

Figure 1: The sample space for rolling two dice.

From Figure 1, you can see that since there is one way to get a sum of two and 36 equally likelyoutcomes, the probability of rolling a two is 1

36 . You can also see that there are six ways to rolla sum of seven (these show up on the diagonal running from the bottom left to the upper right).Therefore, the probability of rolling a seven is 6

36 = 16 . (Notice that getting a total of 7 when the

gray die comes up 3 and the white die comes up 4 is a different outcome from the gray die comingup 4 and the white die coming up 3.)

An event can be more complicated than rolling a given sum. For example, another possibleevent is rolling a sum that is less than 4. To find the probability of this event, we use Figure 1to count the number of outcomes that give a sum which is less than 4. Since there are 3 of these(rolling two ones, rolling a one and a two, or rolling a two and a one), the probability of the eventis 3

36 = 112 .

Oftentimes we can simplify probability calculations by using what is called the complementof an event. Given any event, the complement is defined to be the set of all outcomes in the samplespace that are not in the event. For example, the complement of rolling a sum that is less than 4is rolling a sum that is 4 or greater. Since the sample space has 36 elements and the event “rollinga sum less than 4” has 3 elements, the complement of that event will have 36− 3 = 33 elements.So, the probability of rolling a sum that is 4 or larger is 33/36. In general, if we let E denote an


event, E′ will denote the complement of that event.Since an event E and its complement E′ include every outcome in the entire sample space

with no overlap, we have P (E) + P (E′) = 1 or equivalently P (E′) = 1 − P (E). For example,if E is the event “ rolling a sum less than 4,” then E′ is “rolling a sum 4 or greater.” We haveP (E′) = 1− P (E) = 1− 3/36 = 33/36 or 11/12, just as we calculated above.

To summarize, we have two basic properties of probability:

• The probability of any event, P (E), is a number between 0 and 1 (i.e. 0 ≤ P (E) ≤ 1).

• For any event, E, and its complement, E′, we have P (E) + P (E′) = 1. This means thatP (E′) = 1− P (E).

Permutations and Combinations

The odometer method or multiplication principle works in general for any set of independentchoices. Two special cases based on the multiplication principle are the counting of permutationsand combinations. In both of these, we are looking at choosing a set of r items from a larger setof n items. The basic difference is that in permutations, there is an order or some other way ofdistinguishing between the r items, whereas in combinations, there is no order; we only care whichr items we get.

Permutations

Let’s say the campus Amnesty International chapter has 5 members, who we will list as{A,B,C, D,E} for short. The club is required by student organization by-laws to elect a President,Secretary, and Treasurer. No person can hold more than one position. How many ways is there todo this?

What the club needs to do is an example of selecting a permutation. They are selecting r = 3members from the set of all n = 5 club members, and there is an “order” to the selection. Oneof the three selected will by President, one will be Secretary, and one will be Treasurer. If A isPresident, B Secretary, and C Treasurer this is a different permutation than if B is President, A isSecretary, and C is Treasurer.

To calculate the number of possible permutations, we can use the multiplication principle (orequivalently, the odometer method). We have three choices to make, President, Secretary, andTreasurer. If we choose them in that order, we have 5 ways to choose the President, then 4 ways tochoose the Secretary (since the President cannot also be the Secretary according to the by-laws),then 3 ways to choose the Treasurer. This gives us a total of 5 · 4 · 3 = 60 ways to choose the clubofficers.

What if we did allow a person to hold more than one office?

Well, we would no longer call this a permutation, since we might be choosing 1, or 2, or3 different people to be officers. However, we could still use the multiplication principle. Thenumber of ways of doing this would be 5 · 5 · 5 = 125.

If we listed all 125 ways of choosing the 3 officers allowing a person to hold more than oneoffice, we might start AAA, AAB, AAC, etc. This list would include all 60 of the permutations wehad previously calculated.

Combinations

Now, let us suppose the 5 member club wants to select a homecoming committee of 3 people.Here, there is to be no distinguishing between the 3 people chose. There is no President, Secretary,and Treasurer, just 3 people with the same role as committee members. This is an example of acombination.

We can find the number of these combinations by using our previous calculation of the numberof permutations. One combination would be the set {A, B,C}. These 3 people could also have

12. Probability 249

been chosen as President, Secretary and Treasurer. How many different times would this set ofthree people appear in our list of 60 ways to choose President, Secretary and Treasurer?

Well, if we know A, B, and C are going to be the people chosen, then any 3 of them could bethe President. Again, assuming no doubling up on offices, any of the other 2 could be Secretary,and then the one remaining would have to be Treasurer. This means there are 3 · 2 · 1 = 6 waysthese 3 people could have been chosen for the 3 offices. Thus, these 3 people appear 6 times in thelist of 60.

Now, the same would be true for any other list of 3 people ({A,C, D} or {B, D,E}, etc.). Ineach case, any set of 3 people appears 6 times in the list of 60. So, if we want to count how manydifferent sets of 3 people there are, there must be 60/6 = 10 such sets. We can even list them asfollows:

{A, B, C} {A, D, E}{A, B, D} {B, C, D}{A, B, E} {B, C, E}{A, C, D} {B, D, E}{A, C, E} {C, D, E}

Calculational Shortcuts

Although we have illustrated the thinking behind calculating the number of permutations andcombinations using the multiplication principle in the previous examples, in practice we will useour calculators to find these values. First, for any nonnegative integer n, we define n!, called nfactorial, by

n! = n · (n− 1) · (n− 2) · · · 3 · 2 · 1So, for example, 3! = 3 · 2 · 1 = 6 and 4! = 4 · 3 · 2 · 1 = 24. We can then use factorials to calculatethe number of permutations and combinations as follows:

• The number of permutations of r items selected from a set of n items, denoted P (n, r), is

P (n, r) =n!

(n− r)!.

• The number of combinations of r items selected from a set of n items, denoted C(n, r), is

C(n, r) =n!

r!(n− r)!=

P (n, r)r!

.

Going back to our club example, the number of ways of picking the 3 officers from the club of 5 is

P (5, 3) =5!

(5− 3)!=

5!2!

=5 · 4 · 3 · 2 · 1

2 · 1 = 5 · 4 · 3 = 60.

The number of ways of choosing the 3-person homecoming committee is

C(5, 3) =5!

3!(5− 3)!=

5!3!2!

=5 · 4 · 3 · 2 · 13 · 2 · 1 · 2 · 1 =

12012

= 10.

Even better news, most calculators have buttons for P (n, r) and C(n, r), or have these builtinto a menu. For example, to find these on a TI graphing calculator, hit the MATH menu, andthen choose PRB (short for probability), and you will find P (n, r), C(n, r) and n!. To calculateP (5, 3), go to the main screen and enter the 5, then select the P (n, r) from the menu, then type 3and ENTER. You should get 60.


Thus, your main task in using permutations and combinations is not computational. Theimportant point will be to identify if the situation your are considering is either a permutation ora combination or neither. Remember that permutations and combinations are only special casesof all the possible counting problems that one might run across. To decide if the situation you arelooking at is one of these, you might ask yourself the following:

1. Does this situation involve choosing a set of r items from a larger set of n items?

2. Is there or is there not an “order” or some other distinction being made between the ritems being chosen. If so, you are looking at a permutation. If not, you are looking at acombination.

Probability Distributions

A finite probability distribution is the collection of all the outcomes of a random phenomenontogether with their associated probabilities. For example, a probability distribution for rolling twodice and recording the sum is shown in Table 3. Recall that these probabilities are theoreticalprobabilities, assuming fair dice and equally likely outcomes.

Table 3: The probability distribution for rolling two dice and recording the sum.

Outcome: Sum 2 3 4 5 6 7 8 9 10 11 12Probability 1

36236

336

436

536

636

536

436

336

236

136

The probability distribution shown in Table 4 lists the percentages for various religious pref-erences among 1,384,486 U.S. Armed Forces personnel, as of March 2005.134 In one sense, theseare simply percentages. However, if we consider the random experiment of selecting a U.S. serviceperson and determining their religious preference, these percentages serve as empirical probabilities.

Table 4: The probability distribution for Religious Preferences of U.S. Armed Forces Personnel.

Denomination Name Total Pct.Roman Catholic Church 21.5%No Religious Preference 19.6%Baptist churches 14.7%Christian, no denominational preference 14.0%Unknown 8.1%Protestant, no denominational preference 3.3%Methodist churches 2.6%Lutheran churches 2.4%Southern Baptist Convention 1.3%Church of Jesus Christ of Latter Day Saints (Mormon) 1.3%All Other 11.0%

Note that the outcomes in Table 3 are represented by numbers, but the outcomes in Table 4are religious preferences that have no numerical value. For the dice probabilities, it would make

134These data are compiled by the United States Department of Defense, and were accessed athttp://speakingoffaith.publicradio.org/programs/servingallah/index.shtml

12. Probability 251

sense to ask what the “average roll” would be. For the religious preferences, the best we could dofor an “average” would be to pick the denomination that had the highest percentage (the so calledmode).

For probability distributions which have numerical outcomes, we calculate the mean of aprobability distribution as follows. Multiply each outcome by the respective probability andadd the products. For example, the mean of the probability distribution in Table 3 is

2 · 136

+ 3 · 236

+ 4 · 336

+ 5 · 436

+ 6 · 536

+ 7 · 636

+ 8 · 536

+ 9 · 436

+ 10 · 336

+ 11 · 236

+ 12 · 136

= 7.

This means that, although we cannot say what will happen on any particular roll, in the long runthe average of our rolls should be 7 or at least close to 7.

Coin Flipping Distributions

Suppose three coins are flipped and we are interested in the event of getting a certain numberof tails. Using the odometer method, we can think of each coin as a slot, with two choices (H or T)for each slot. Thus, there are 2 · 2 · 2 = 8 possible outcomes, which we can easily list: HHH, HHT,HTH, THH, HTT, THT, TTH, TTT. If it helps, we can think of the coins as a penny, a nickel,and a dime, so heads on the penny and the nickel and tails on the dime – HHT – is different fromheads on the penny and dime and tails on the nickel – HTH.

From the list of elements in this sample space, we can see that there is 1 way to get 0 tails, 3ways to get 1 tail, 3 ways to get 2 tails and 1 way to get 3 tails. The probability distribution forflipping 3 coins and recording the number of tails is shown in Table 5.

Table 5: The probability distribution for the number of tails observed when tossing 3 coins.

Number of Tails 0 1 2 3Probability 1

838

38

18

To find the mean of this probability distribution, multiply each outcome times its correspondingprobability and then add, just as we did with the dice example.

mean number of tails = 0(

18

)+ 1

(38

)+ 2

(38

)+ 3

(18

)=

08

+38

+68

+38

=128

= 1.5

Alternatively, we could calculate the mean by considering the the eight different outcomes, addingup the number of tails for each outcome, and dividing by 8.

mean number of tails = (0 + 1 + 1 + 1 + 2 + 2 + 2 + 3)÷ 8 =128

= 1.5

Notice we get the same answer either way. The first method is more efficient, and also makes thedivision by 8 part of the probability instead of an operation at the end.

Have we seen this idea and these types of sums before?

Yes, in fact, we have. If you go back to our discussion of GPA in Chapter Four, you will seethat the way we calculated GPA is very much like what we are doing here with the mean of aprobability distribution. Now, grades are not typically assigned at random (at least we hope not),but the idea of mean or average in both these situations is similar. In fact, if all the grades atyour college were assigned at random, but according to some set percentages for each grade (10%of students in any class get an A, 25% get a B, etc.), then the mean of the probability distributionwould represent the expected GPA of students at your college!


Lotteries

Lotteries of all sorts have become almost ubiquitous, and many state budgets depend heavilyon lottery revenue. Let’s consider one example from the state of Colorado. In one game, simplycalled Lotto, the player picks six numbers from the set of integers 1 - 42. The state then picks six“winning numbers.” The players win prizes by matching some or all of the six winning numbers.The probability distribution where the outcomes are the number of matches with their associatedprobabilities is shown in Table 6.

Table 6: The probability distribution for the number of matches in the Colorado Lotto.

No. of matches 6 5 4 3 2 1 0Probability 1

5,245,786216

5,245,7869,450

5,245,786142,800

5,245,786883,575

5,245,7862,261,9525,245,786

1,947,7925,245,786

As Decimal 1.906× 10−7 4.118× 10−5 0.0018 0.0272 .1684 .4312 .3713

Notice that the probability of matching all six numbers is 15,245,786 . How bad are these odds?

Imagine yourself in a long line of 5,245,786 people waiting to buy a single lotto ticket, and that oneperson in that line will be the winner. Such a line would go roughly from South Dakota to NewYork City! Your probability of winning is the same as the probability that you would get pickedout of that line at random.

To find out how many numbers you would expect to match on average if you bought a ticket,we find the mean of the distribution using given in Table 6:

6(

15, 245, 786

)+ 5

(216

5, 245, 786

)+ 4

(9, 450

5, 245, 786

)+ 3

(142, 800

5, 245, 786

)+ 2

(883, 575

5, 245, 786

)

+1(

2, 261, 9525, 245, 786

)+ 0

(1, 947, 7925, 245, 786

)≈ 0.86

This means that, on average, you will match 0.86 of the winning numbers. In other words, youcan expect to match less than one number, on the average. Therefore, it should not surprise youif you buy a Lotto ticket and do not match a single number.

Of course, players are less interested in how many numbers they match than in how muchmoney they win. We could calculate the money won, on average, from playing lotto by replacingthe number of matches with the prizes one in Table 7. In this case, the mean is also called theexpected value of the lottery. The jackpot starts at $1 million dollars, and increases each time noone matches all six numbers. The other prizes also vary somewhat. The probability distributionas of August First, 2007 was as is shown in Table 7.

Table 7: The probability distribution for prize winnings in Colorado Lotto.

No. of Matches 6 5 4 3 2 1 0Winnings $1,919,961 $516 $41 $3 $0 $ 0 $0

Probability 15,245,786

2165,245,786

9,4505,245,786

142,8005,245,786

883,5755,245,786

2,261,9525,245,786

1,947,7925,245,786

The expected value of the lottery, or in other words the mean of this probability distribution,is calculated as follows.

1, 919, 961(

15, 245, 786

)+516

(216

5, 245, 786

)+41

(9, 450

5, 245, 786

)+3

(142, 800

5, 245, 786

)+0

(883, 575

5, 245, 786

)

12. Probability 253

+0(

2, 261, 9525, 245, 786

)+ 0

(1, 947, 7925, 245, 786

)≈ 0.543

So, the expected value for playing the Colorado Lotto is 54 cents. This means that you couldexpect to win, on average, just 54 cents per ticket. Because it costs $1 for each ticket, you lose46 cents per ticket on average. So, if you bought 100 tickets, costing you $100 total, you shouldexpect to win only about $54. This is not a particularly good deal for the overwhelming majorityof lotto players, but it certainly helps fill the state’s coffers.

Conditional Probability

DNA fingerprinting and other probability calculations used in court cases, medical testing fordiseases, and many other examples of probabilities that one reads about are examples of what areknown as conditional probabilities. Almost anytime a probability is qualified by some sort of“condition,” it is a conditional probability.

For example, in 1998, the percent of all live births that were classified as low birthweight was7.8%.135 If we consider this number as a probability, we could say the probability that a randomlyselected baby born in 1998 was of low birthweight was 7.8% (or 78 out of 1000 if you prefer relativefrequencies).

However, for mothers who smoked, the percentage was 12.0%. We could say the probabilitythat a randomly selected baby born in 1998 was of low birthweight, given that the mother wasa smoker, was 12%. This is a conditional probability, with the condition being “given that themother was a smoker.”

Conditional probabilities can be a little cumbersome to write out in words. A shorthand wayof denoting the probability that a given baby is low weight given that the mother smokes is

P (low weight|mother smokes).

So, we could write P (low weight) = 0.078, while P (low weight|mother smokes) = 0.12. Theevent we are interested in appears in front of the vertical line, and the condition appears after.

You might guess that both the conditional and the “unconditional” probabilities in this caseare empirical probabilities. The 7.8% is the result of taking the total number of low weight birthsdivided by the total number of births. The conditional probability of 12% was computed by dividingthe number of low weight births where the mother was a smoker divided by the total number ofbirths where the mother was a smoker. The difference is, for the conditional probability, we lookat the same event, but restrict ourselves to the sample space given by the stated condition. In thisfirst example the condition is “mother was a smoker.”

Figure 2 gives a Venn diagram that might be useful in picturing this situation.P (low weight) is the number of births represented by the “low weight births” circle divided by

the total number of births, represented by the rectangle. The conditional probability

P (low weight|mother smokes)

is calculated using only the “births to smoking moms” region. The numerator would be the numberof births in the overlap of “low weight births” and ’“births to smoking moms” divided by the numberof births in the ’“births to smoking moms” region.

In many cases, one could switch the event and the condition around and still have a probabilitythat makes sense. For example, we might be interested in

P (mother smokes|baby is low weight).

This would be computed by dividing the total number of low weight babies born to motherswho smoke, by the total number of low weight babies. In the Venn diagram, the sample space forthis probability would be represented by the “low weight births” circle.

How is P (mother smokes|baby is low weight) different than P (low weight|mother smokes)? Notethat, in both cases, the numerator of the probability fraction would be the number of low weight

135This and the following data are from The Statistical Abstract of the United States: 2001.


Births to Smoking Moms

Low Wt. Births

All Births

Figure 2: Venn diagram showing possible set relations for births example

babies born to mothers who smoke. What is different is the denominators of the probability frac-tions. In the first case, the denominator is the number of low weight babies, and in the secondcase the denominator is the number of babies born to mothers who smoke. These are two differentgroups of babies, but both of these groups contain the group counted in the numerator, namelythe babies who are low weight and who are born to mothers who smoke.

Example 1. Suppose that, at a particular hospital, there were 25 births last month, and6 of these were to women who smoked. Suppose that 1 of the six babies born to the smokerswas low weight and 2 of the other babies was low weight. Find P (low weight|mother smokes)and P (mother smokes|baby is low weight). Also, find P (low weight|mother does not smoke) andP (mother does not smoke|baby is low weight).

Solution: For both P (low weight|mother smokes) and P (mother smokes|baby is low weight),the numerator is the number of low weight births to mothers who smoke, which is 1. We have

P (low weight|mother smokes) =16≈ 17%.

P (mother smokes|baby is low weight) =13≈ 33%.

For the other two probabilities, the numerator is 2, since there were two low weight babies bornto mothers who do not smoke. The probabilities are

P (low weight|mother does not smoke) =219≈ 10.5%.

P (mother does not smoke|baby is low weight) =23≈ 67%.

Example 2. How do you think newspapers or television stations might report the situationwith regards to low weight births at the hospital in this last example? Here are a few possiblesample headlines (you might create some more of your own just for fun).

1. Local data show smoking increases chance of low weight birth 6.5%

2. Local data show smoking increases chance of low weight birth 62%

3. Smoking accounts for one third of all low weight births

12. Probability 255

4. Low weight babies twice as likely to be born to non-smokers

5. Babies and smoking: Is there anything wrong?

Hard to believe all these could be based on the same information! Explain the calculations andthinking behind each of these headlines. Which of them do you think is the most “accurate?”

Solution: Let’s go through these each in turn.

1. We can get the 6.5% by subtracting the 10.5% probability of low weight birth given the motherdoes not smoke from the 17% probability for smokers. Is this a valid way to compare?

One might make the case that this is not telling the whole story. After all, you would get thesame 6.5% difference whether the given probabilities were 7% and 0.5%, or 56.5% and 50%, orthe actual probabilities reported above. If the two probabilities are low the effects of smokingare very great, while if the two probabilities are high, the relative effect of smoking seems tobe low. The main problem is that we don’t know what the 6.5% compares to without havingall the information.

2. This one was calculated by taking the 6.5% and dividing by the 10.5%. The advantage tothis approach is that it gives you a sense of the relative effect of smoking versus non-smoking.The 62% calculated this way does tell you there really is a 62% greater chance of having alow weight baby if you smoke versus if you do not smoke. The headline worded this way istechnically accurate.

There is one very legitimate criticism that could be made with this approach, however. Itdoes not take into account whether having a low weight baby is a very rare or a relativelycommon occurrence. In our example, there were 3 low weight births out of a total of 25 birthswhich is a 12% overall rate. Suppose instead we had 10,000 smokers and 10,000 nonsmokers,and that there were 13 low weight babies among the smokers and 8 among the non-smokers.The difference is 5 low weight babies and 5/8 does give you 62%. We could say, accurately,that smoking increases your chances of a low weight birth by 62%. However, out of 10,000smoking moms, you are only going to get 5 more low weight births than in the nonsmokinggroup. Five out of 10,000 is a very small percentage. So, even though the percentage increasein the risk is great (62%), the event is so rare that it might not be worth worrying about.The moral is that percent increases like this are more important the more common the eventis.

3. The 33% is calculated by using the number of low weight births. There were 3 low weightbirths and one of them was to a smoking mom.

This percentage is very misleading, and really does not provide much information by itself.The fact that one out of three low weight babies were born to smoking moms does not tellyou anything unless you know what percentage of moms overall were smokers. In our case,we have 25 moms and 6 were smokers. Six out of 25 is 24%, and this is the percent of allmoms who smoke. Knowing that this percentage (24%) is smaller than the 33% does tell youthat smoking is correlated with an increased risk of having a low weight birth. Wheneveryou see a percentage like this, based on a sub-population, you should look for the percentagefor the entire population. If this is not given, you really have information only about the onesubgroup, in this case the smokers. You do not have any information about how the smokinggroups compares with the nonsmoking group.

4. This is also based on the two low weight babies born to nonsmokers and the one born tosmokers. While it is true that twice as many low weight babies were born to nonsmokers, theheadline ignores that there are more than three times as many nonsmokers overall as smokersoverall. As in the last item, you need to know the relative sizes of the larger populations.This headline would only be accurate if the number of smoking moms was the same as thenumber of nonsmoking moms.


5. We won’t say too much about this one. This headline might be be the result of the same badthinking that went into the last one.

So, what headline should we use? This author would humbly suggest something generic like“local hospital releases birth statistics.” The details could then be a part of the article where theycan be adequately explained.

Multi-stage Experiments

We have seen that determining the probability of an event by first writing down all the outcomesof the sample space and then counting the number of outcomes in the event works well when therandom phenomenon is fairly simple and the number of outcomes is fairly small. When eventsbecome more complex, we used the multiplication principle to count the sample space and thenumber of outcomes in any event. In general, we have been calculating the numerator first, possiblyby multiplying, and then dividing by the total number of possible outcomes. However, sometimesit is useful to first calculate individual probabilities, and then multiply to get the probability of anevent in a multi-step process. Corresponding to our multiplication rule for counting, we have thefollowing property of probability:

The probability of two or more independent events is the product of the associated prob-abilities.

For example, suppose we roll a die twice and want to calculate the probability of rolling a 2 onthe first roll and an even number on the second roll. We could look at the sample space in Figure1 and count the number of outcomes for which this is true. If we think of the number on the leftin each pair as the outcome for the first roll and the number on the right as the outcome for thesecond roll, we see that there are three outcomes in which the first roll is a 2 and the second is even.These are (2,2), (2,4), and (2,6). Therefore, the probability of this event happening is 3

36 = 112 .

On the other hand, we could compute this probability using our multiplication rule. Theprobability of rolling a 2 on the first roll is 1

6 . The probability of rolling an even number on thesecond roll is 3

6 since there are 3 even numbers on a die. Since the outcome of the first die doesnot effect the outcome of the second, the two events are independent. So, we can multiply theprobabilities of these two events to determine the probability of both events happening. Therefore,the probability of rolling a 2 on the first roll and an even number on the second is 1

6 · 36 = 3

36 = 112 .

Observe that 16 of the outcomes in Figure 1 have a 2 on the first die and 3

6 = 12 of the outcomes in

Figure 1 have an even number on the second die. Therefore, 16 of 1

2 is 16 · 1

2 = 112 .

It is very important to remember that this multiplication rule for probabilities only works ifthe events are independent, just as with the multiplication rule for counting. Remember that twoevents are independent if the occurrence of one does not change the probability of the otheroccurring.

Not all events are independent. For example, the probability of you having blond hair isdependent on whether or not either of your parents has blond hair. You are more likely to haveblond hair if one or both of your parents does. With dice, the events “rolling an odd number” and“rolling a prime number” are dependent. If you roll an odd number, 1, 3, or 5, you have a 2/3probability of rolling a prime number. If you roll an even number, 2, 4, or 6, you only have a 1/3chance of rolling a prime number.

The probability that we may fail in the struggle ought not to deter us from the supportof a cause we believe to be just.—Abraham Lincoln

Reading Questions for Probability

We encourage children to read for enjoyment, yet we never encourage them to “math”for enjoyment. We teach kids that math is done fast, done only one way and if youdon’t get the answer right, there’s something wrong with you. You would never teachreading this way.—Rachel McAnallen, from Math? No Problem, The Hartford Courant, October, 1998.

1. For each of the following probabilities, say whether the probability is empirical, theoretical,or subjective and why you think so. Numerical values are not given for these probabilities,and you need not do any calculations. Your answer should be based on how you think theprobability is ‘probably’ calculated.

(a) The probability of rolling doubles when rolling two dice.(b) The probability that a given U.S. citizen died from West Nile Virus during the last 30

days.(c) The probability that a given U.S. citizen will die from West Nile Virus during the next

30 days.(d) The probability that a Republican will win the next presidential election.(e) The probability that you will win the jackpot in the next powerball lottery drawing.(f) The probability that you will be selected for jury duty during the coming year.(g) The probability that it will rain tomorrow.

2. For each of the following lists or descriptions of possible outcomes, say whether the outcomesin the list are equally likely or not, and why you think so.

(a) The outcomes are {heads, tails} when flipping one fair coin.(b) The outcomes are all the possible combinations of numbers in the powerball lottery.(c) The outcomes are the numbers {2,3,4,5,6,7,8,9,10,11,12} that are the sums you can get

when rolling two fair dice.(d) The three outcomes are {two heads, one head and one tail, two tails}, when flipping two

fair coins.(e) The outcomes are all the possible numbers one could get by counting how many U.S.

citizens vote in the next presidential election.

3. For each pair of events, say whether you think the events are independent or not, and whyyou think so.

(a) You roll two fair dice. The two events are rolling a 5 on the first die, and rolling an evennumber on the second die.

(b) You roll two fair dice. The two events are rolling a 6 on the first dice, and rolling a sumless than 9.

(c) You select one U.S. citizen at random. The two events are the person has a collegedegree, and the person had an annual income of more than $40,000 during the last fullcalendar year.

4. Suppose a new car model comes with the following option: 3 choices of body style (2-door,4-door or station wagon), 4 colors (black, red, green or white), and two choices of engine size(4 or 6 cylinder). List all the different ways the model could be ordered starting with (2,B,4)for a 2-door black 4-cylinder model.

5. How many different lunch choices would there be if you choose from 4 sandwiches, 2 sideorders, and 5 drinks? [Note: You do not need to list all of the possible combinations.]

257


6. Why can a probability never be greater than 1?

7. If the probability it will rain tomorrow is 0.6, what is the probability of the complement ofthat event?

8. Suppose you are tossing 3 coins and want at least 2 heads.

(a) List all of the possible outcomes for the sample space.(b) List all of the possible outcomes for the event.(c) What is the probability of tossing 3 coins and having at least 2 heads?

9. You roll two fair dice.

(a) What is the probability that the sum is 7.(b) What is the probability that the sum is greater than or equal to 7, but less than 10?(c) What is the probability that the sum is 5 or 6?(d) What is the probability that the sum is less than 6, or a prime number?

10. Suppose you have a jar with 5 white marbles and 8 black marbles.

(a) What is the probability of drawing a black marble?(b) What is the probability of drawing a white marble?(c) Find the sum of your answers from parts (a) and (b). Why is this true?(d) Suppose you draw one marble, replace it, and draw a second marble.

i. What is the probability that both marble are black?ii. What is the probability that both marbles are white?iii. What is the probability that the first marble is black and the second one is white?

11. When rolling two dice, what is the probability that you get a three on the first die and anumber greater than 3 on the second?

12. Calculate the following

(a) C(5, 2)(b) P (6, 3)(c) C(6, 3)(d) C(25, 10)

13. Suppose the local College Republican group consists of 9 members.

(a) How many ways can the club choose 4 officers, President, VP, Secretary and Treasurer?(b) How many ways can the club choose a publicity committee of 4 people?(c) How many ways can the club choose a publicity committee of 5 people?(d) Why is the answer to the last two questions the same!?(e) Suppose the club has 5 men and 4 women. How many ways is there to choose a pub-

licity committee of 4 people if they want to have equal gender representation on thecommittee?

14. What is a probability distribution?

15. Give an example, not mentioned in the reading, of a probability distribution where it ispossible to find the mean.

12. Probability 259

16. Give an example, not mentioned in the reading, of a probability distribution where it is notpossible to find the mean.

17. List the sample space for the experiment in which four coins are flipped. (Hint: There are16 different outcomes.)

18. When flipping four coins, determine the following probabilities.

(a) Getting four heads.(b) Getting at least 3 heads.(c) Getting a head on the second coin.

19. Find the mean of the following probability distribution where the outcome is number ofpeople in a family. Describe what your answer means.

Persons in Family 2 3 4 5 6 7Probability 0.41 0.24 0.21 0.09 0.03 0.02

20. Aunt Hazel’s Raisin Cookies were analyzed for their number of raisins. The following prob-abilities were found:

Number of Raisins, x 3 4 5 6 7 8Probability of x Raisins 0.10 0.15 0.15 0.25 0.20 0.15

Find the mean number of raisins in a cookie.

21. Calvin is taking a test using the “Calvin Method” which involves flipping a coin and putting“true” for heads and “false” for tails. There are ten questions on the test

(a) What is the probability that Calvin gets all ten questions right?(b) What is the probability that Calvin gets nine out of 10 correct? (Hint: there are C(10, 9)

ways to pick which 9 questions will be the correct ones).(c) What is the probability that Calvin gets exactly 8 of 10 questions correct?

22. A family has three children.

(a) What is the probability that the two oldest are girls and the youngest is a boy?(b) What is the probability that there are 2 girls and one boy (in any order)?

23. Suppose that, when throwing darts, you make a bull’s eye 40% of the time. If you throw 2darts, what is the probability that none of them is a bull’s eye?

24. You and a friend decide to play “gambling dice.” It costs 1 dollar to play. If you roll a 5 or6, you get 2 dollars (a 1 dollar profit). If not, you lose your dollar. Compute the expectedvalue for this game.

25. You have taken a survey of 235 college students. The results are:

• 67 students smoke cigarettes. Call this group A.• 18 students use chewing tobacco. Call this group B.• 103 students reported having one or both parents smoked. Call this group C.• 46 students are in both A and C.• 6 students are in both A and B.


• 9 students are in both B and C.• 4 students are in all three groups, A, B, and C.

(a) Draw a Venn Diagram, including circles for A, B, and C, and fill in the number ofstudents in each region of the diagram.

(b) What is the probability that a student smokes, given their parents smoked.(c) What is the probability that a student’s parents smoked, given that the student smokes.(d) Are students who smoke more or less likely to use chewing tobacco than students who

don’t smoke? Explain.(e) Are students whose parents smoked more or less likely to use chewing tobacco? Explain.

Probability: Activities and Class Exercises

1. Medical Testing.

Mr. Speaker, once again, the United States has an opportunity, and the respon-sibility, to lead the world in confronting one of the most compelling humanitarianand moral challenges facing us today. I speak of the HIV/AIDS pandemic, a crisisunparalleled in modern times and one that threatens the entire world, embracingboth developed and developing countries alike.—Representative Henry Hyde, U.S. House of Representatives, December 11, 2001

One of the tests that is available to determine if someone has the human immunodeficiencyvirus (HIV) is called the Single Use Diagnostic System (SUDS). This test will give a positiveresult in approximately 97.9% of the cases where someone has the virus and 5.5% of thosewho do not have the virus.136 The latter is called a false positive.Suppose you are living in a city of 100,000 people. It is estimated that approximately 0.3%of the United States population is infected with HIV.137 We will assume that this percentageis true for the hypothetical city you are living in as well.Suppose you test positive for HIV using the SUDS test. Since there is a 5.5% chance offalse positives among those without the virus, there is some chance that you do not have thevirus, even though you have tested positive. We want to find the probability that youactually have the virus assuming you have tested positive. To do this, you need toknow how many people in your city tested positive, and how many of this group actuallyhave the virus.

(a) Let’s first consider those people who have the virus.i. What is the estimate for how many people in your city have HIV?ii. Suppose all those who have the virus in your city were tested. Approximately how

many would test positive and how many would test negative?(b) Suppose everyone who did not have the virus were tested.

i. How many people in your city do not have HIV?ii. Of these, how many would you expect to test positive, and how many would test

negative?(c) Using parts (a) and (b), if everyone in your city were tested for HIV, both those with the

virus and those without, how many would test positive? How many would test negative?(d) You have tested positive for HIV. What is the probability that you actually have the

virus? Explain how you arrived at your answer.(e) Does your answer in part (d) seem surprising? Explain why the probability is so much

lower than the 97.9% accuracy for those having the virus and the 94.5% chance thatthose without the virus test negative.

(f) One erroneous result is a false positive. Another is a false negative. Given the accuracyof the test and the other data given, what is the probability that a person who testsnegative for the virus actually has HIV?

136Centers for Disease Control and Prevention, “Rapid HIV Testing: 2003 Update,” 28 October 2003,<http://www.cdc.gov/hiv/rapid testing/materials/ USCA Branson.pdf>.

137Centers for Disease Control and Prevention, 28 October 2003, “HIV/AIDS Update,”<http://www.cdc.gov/nchstp/od/news/At-a-Glance.pdf>.

261


2. Relative RisksThe following passage is from a report by the U.S. State Department, cited in an article fromthe U.S. Embassy in Tokyo, Japan. (http://japan.usembassy.gov/e/p/tp-20030501a9.html).

Washington – A decline in international terrorist attacks – down 44 percent in2002 from the previous year – can be attributed to an intensively waged war onterrorism across every region of the world, the U.S. State Department’s annualreport on terrorism says.International terrorists conducted 199 attacks in 2002, compared with 355 attacksin 2001, according to the annual “Patterns of Global Terrorism: 2002” report,released April 30. Over the same period, the number of people killed by terroristsdeclined to 725, down dramatically from the 3,295 people killed in 2001. The 2001total included those killed in the September 11 attacks in New York, Washingtonand Pennsylvania. The deaths in 2002 included 30 U.S. citizens.By geographic region, there were five terrorist attacks in Africa, 99 in Asia, sevenin Eurasia, 50 in Latin America, 29 in the Middle East, none in the United States,and nine in Western Europe, the report said.

(a) One question we could ask is “who is more likely to be a victim of terrorism, a U.S.citizen, or someone who is not a U.S. citizen?” Note that the world population in 2002was approximately 6.23 billion. The U.S. population was approximately 288 million, or.288 billion.

i. What is the probability that a person who was a U.S. citizen was killed by terroristsin 2002?

ii. What is the probability that a person who was not a U.S. citizen was killed byterrorists in 2002?

iii. Based on your previous two answers, who was more at risk of being killed by ter-rorism in 2002, U.S. citizens or people who were not U.S. citizens?

(b) Another way to answer the previous question is to consider whether the percentage ofthose killed by terrorists who were U.S. citizens was larger or smaller than the percentageof the overall population that is represented by U.S. citizens.

i. What is the probability that a person killed by terrorism in 2002 was a U.S. citizen?ii. What is the probability that a given person is a U.S. citizen?iii. Which of these two numbers are larger and what does this tell you about the risk

to U.S. citizens?iv. Explain why “the probability that a person killed by terrorism in 2002 was a U.S.

citizen” is not the same as the “probability that a person who was a U.S. citizenwas killed by terrorists in 2002?”

(c) One could criticize the way we structured the question in part (a) since it disregardsthe facts in the third paragraph from the Embassy report. Note that not a single(international) terrorist attack occurred in the U.S. in 2002. How does this change theconclusion or the reasoning you used in part (a)? What might this imply about whichU.S. citizens are at risk from terrorist attacks?

(d) One way to measure how concerned one should be about particular risks is to comparerisks from different sources. From (a) and (b), you have some information about the riskof terrorism. Let’s consider some other sources of risk. Just on the basis of your owngut intuition, rank the following risks of death from most likely to least likely. Assumewe are only considering the U.S.

i. Lightningii. Airline crashiii. Automobile crashiv. Anthrax poisoning

12. Probability 263

v. Terrorism in generalvi. Murder in general

(e) For comparison, here are some facts on these risks.

• The number of people killed every year in car accidents in the U.S. is more than40,000.

• On average, about 90 people a year are killed by lightning.• According to the FBI, 15,517 people were murdered in 2000.• Over the last 19 years, on average about 120 people have died in airline crashes each

year.• In 2001, 5 people died from Anthrax poisoning.

These are not all from 2002, but all are from recent years. Calculate the probability ofdying from each of the risks listed above, ranking them from most likely to least likely.Instead of using decimals or percents, give the probabilities as a ratio “1 out of N”.

(f) How many times greater is the probability of the most likely risk than the probabilityof being killed by terrorists?

(g) It might be worth comparing any of these to the probability of winning the jackpot in oneof the popular powerball lotteries. In one such lottery, there is a 1 in 54 million chance ofwinning the jackpot. (Data from http://www.unitedjustice.com/stories/stats.html andhttp://www.worldwatch.org/pubs/mag/2000/135/mos/). Which of the above risks aremore likely than winning this lottery, if any?

(h) Worldwide, the United Nations Food and Agricultural Organization estimates thatabout 25,000 people die every day from starvation. Given this, about what is theprobability that a randomly selected person died from starvation in 2002? How doesthis compare to some of the other risks you’ve considered?

3. The 2008 election. In any election, politicians are most concerned with the most basicquestion; who wins? However, exit polls which indicate how different demographic groupsvoted are also of great interest. In this activity, we consider exit polling data from thispast fall, accessed from http://www.cnn.com/ELECTION/2008/results/polls.main/ onDecember 30th of 2008.

Table 8: Results by Race.

Total Obama McCain OtherWhite (74%) 43% 55% 2%African-Amer. (13%) 95% 4% 1%Latino (9%) 67% 31% 2%Asian (2%) 62% 35% 3%

(a) Based on the table above, what is the probability a given voter in the last election waswhite? African-American?

(b) Based on the table, what is the probability that a person voted for Obama given thatthey were white? African-American? Asian?

(c) Since a Presidential election is really a set of 51 state elections (including WashingtonD.C.), we cannot say who “would have” one the election under different circumstanceswithout looking at those circumstances state by state. However, we can look at theoverall popular vote totals and how they would change if certain demographic groupshad voted differently.


i. According to CNN, 66,882,230 Americans voted for Obama and 58,343,671 votedfor McCain. It is estimated a total of 127,003,956 Americans voted. Based on thesetotals, estimate how many people are represented by each of the 12 categories inthe table. (Hint: You can actually do this using only the total number of votersand the percents given in the table). You may want to simply create another table,giving the total voters in each category instead of the percentages.

ii. Some Obama critics claimed that many African-Americans only voted for Obamabecause of his racial background. One indication of whether this is true or not isto look at how African-Americans voted in previous elections. This is not a perfectmeasure, since many factors can influence voters and the four years between electionscan effect these factors. However, in the 2004 election, democrat John Kerry won88% of the African-American vote to 11% for George Bush. Suppose that, in 2008,the percentages in the table above remained the same except that the percent ofAfrican-Americans who voted for Obama had been 88% instead of 95% and thatMcCain won 11% of the African-American vote. How would this have changed thepopular vote totals?

Appendix 1: Miscellaneous Mathematics Reference

Exponents and Radicals

Exponents are shorthand for repeated multiplication. We define xn to mean

xn = x · x · x · · ·xwhere there are n x’s multiplied together. In particular, x1 = x.Roots are defined as follows. We let n

√x denote the number a so that an = x. In other words

( n√

x)n = a. This definition is subject to the following conditions.

1. If n is even, x must be greater than or equal to 0, and in this case n√

x ≥ 0.

2. If n is odd, there is no condition on x, and n√

x is the same sign as x.

Properties of exponents and radicals

We have the following properties of exponents. The proofs of these properties, in other words,the reason they always hold true, are all based on the definition of exponents given above.

1. xn · xm = xn+m

2. xn

xm = xn−m

3. (xn)m = xmn

4. (xy)n = xnyn

5.(

xy

)n= xn

yn

6. x0 = 1 for all x 6= 0

7. x−n = 1xn

8. xm/n = n√

xm

Because of property 8, properties of radicals can be derived from the properties of exponents,since radicals can always be expressed as exponents.

Logarithms

In Chapter 7, we defined logb x by

logb x = y if and only if by = x

if b = 10, we write log x and if b = e, we write lnx.As noted in Chapter Seven, it is important to keep in mind how logarithms fit into the usual

order of operations. In general, unless there are parentheses involved, logarithms are done afterany multiplications, exponents, or divisions that come immediately after the log sign, but beforeany additions or subtractions and before any multiplications immediately preceding the log sign.

Consider the following examples.

1. To calculate log xy, you would first multiply x and y, and then take the logarithm of theproduct. This is the same as log(xy). On the other hand, to calculate (log x)y, you wouldfirst take the logarithm of x and then multiply the result by y.

265


(a) log 2 · 3 = log 6 ≈ .77815(b) (log 2)3 ≈ .30103 · 3 = .90309

2. x log y is the same as x(log y) or (log y)x. For example, 3 log 2 ≈ 3 · (.30103) = .90309.

3. log xn is equal to log(xn) and not (log x)n.

(a) log 23 = log 8 ≈ .90309, but (log 2)3 ≈ .301033 ≈ .02728.(b) Property 9 below in the properties of logarithms applies to log xn and not (log x)n.

Note that in the previous item, we calculated log 23 ≈ .90309, and that this equals3 log 2 ≈ 3 · (.30103) = .90309.

4. Additions and subtractions come after logarithms, multiplications and exponents, unlessparantheses are involved.

(a) log 3 + log 2 ≈ .47712 + .30103 = .97815(b) log 3 + 2 ≈ .47712 + 2 = 2.47712(c) log(3 + 2) = log 5 ≈ .69897

Properties of the Logarithm Function

In Chapter Seven, we noted the following properties of the logarithm function.

1. log 10x = x.

2. log(1) = 0.

3. log x is negative for 0 < x < 1 and positive for x > 1.

4. y = log x has a domain of x > 0 and a range of all real numbers.

5. The graph of y = log x is increasing and concave down.

6. logb x = log blog x for any positive b 6= 1.

7. log(uv) = log u + log v.

8. log(u/v) = log u− log v.

9. log xn = n log x.

10. 10log x = x.

Let’s prove properties 8 and 9, as promised in the text.First, property 8. Let a = log u and b = log v. So, u = 10a and v = 10b. Then u/v = 10a/10b =

10a−b, using property for exponents above. Taking logarithms of both sides, we get

log(u/v) = log 10a−b = a− b

using property 1 for logarithms above. But, a − b = log u − log v. This implies log(u/v) =log u− log v.

Now for property 9. Let a = log x. So, x = 10a. This implies xn = (10a)n = 10an using property3 for exponents. Since xn = 10an, we have from the definition of logarithms that log xn = an = na,and since a = log x, we get

log xn = na = n log x

Appendix 1: Miscellaneous Mathematics Reference 267

The number e and natural logarithms

We used the common base 10 logarithm above, but the properties and proofs work for all bases.We noted in the text that if e, which is approximately 2.71828, is the base, we write lnx insteadof loge x. Where does e come from?

A rigorous definition of e is given in most calculus books. There, you might find e defined by

e = limx→∞

(1 +

1x

)x

This means, that if we calculate the expression (1 + 1/x) for larger and larger x, the values weget should get closer and closer to one particular number, and we use the letter e to denote thisnumber (so we don’t always have to write out “approximately 2.71828”). The table below gives afew sample calculations.

x (1 + 1/x)1 210 2.59374246100 2.7048138291000 2.716923932

1,000,000 2.718280469

Table 1: Numerical approximations to e

The number e is important because of it’s relation to exponential growth. Recall the compoundinterest formula B(t) = P (1+ i)t, where i is the periodic interest rate and t is the number of years.If we let r be the annual rate, and n be the number of compounding periods per year, then i = r/n,and the number of periods equals n times the number of years, which we’ll denote m. We get,

B(t) = P (1 + r/n)nm

If we let P = 1, r = 1 (denoting 100% interest) and m = 1, then the expression for B(t)becomes exactly the expression used to define e. Letting n get larger and larger would give youbetter and better approximations for e.

Letting n get larger and larger corresponds to having more and more compounding periods peryear. n = 12 is monthly compounding, n = 365 is daily compounding (you get paid interest everyday). For any P , r, and m, we get that

B(t) = P (1 + r/n)nm ≈ Perm

for large n. In other words, if you compound often, you can use the formula Perm as an approxi-mation.

Many types of quantities, other than money in an account, grow exponentially. For example,populations might grow at an approximately fixed percentage growth rate. However, additions tothe population usually happen continuously, not once per year or once per month (except in speciesthat do breed at fixed times of the year). So, the Perm is more appropriate than the compoundingformula, since it represents continuous compounding.

Why didn’t we use e when modeling populations?

Actually, we could have used e. Any model of the form y = abx can be expressed using e byfinding r so that er = b. If we find this r, then abx = a(er)x = aerx. Finding r is not a problem,since if er = b, then r = ln b. We used the y = abx only to avoid the complications of discussing e.

Appendix 2: Data Samples

The effort of the economist is to ”see,” to picture the interplay of economic elements.The more clearly cut these elements appear in his vision, the better; the more elementshe can hold and grasp in his mind, the better. The economic world is a misty region.The first explorers used unaided vision. Mathematics is the lantern by which whatbefore was dimly visible now looms up in firm, bold outlines. The old phantasmagoriadisappear. We see better. We also see further.— Irving Fisher

World Population

The data in these tables is from the United States Census Bureau athttp://www.census.gov/ipc/www/worldpop.html.

Percent Avg. ann. Percent Avg. ann.Year Population grth. rate pop. change Year Population grth. rate pop. change1950 2,555,360,972 1.47 37,785,986 1977 4,231,356,327 1.69 72,172,2861951 2,593,146,958 1.61 42,060,389 1978 4,303,528,613 1.73 75,085,4091952 2,635,207,347 1.71 45,337,232 1979 4,378,614,022 1.71 75,655,1811953 2,680,544,579 1.77 47,971,823 1980 4,454,269,203 1.69 75,864,5641954 2,728,516,402 1.87 51,451,629 1981 4,530,133,767 1.75 80,105,0081955 2,779,968,031 1.89 52,959,308 1982 4,610,238,775 1.73 80,253,7641956 2,832,927,339 1.95 55,827,050 1983 4,690,492,539 1.68 79,312,0071957 2,888,754,389 1.94 56,506,563 1984 4,769,804,546 1.68 80,596,5051958 2,945,260,952 1.76 52,335,100 1985 4,850,401,051 1.68 82,324,4171959 2,997,596,052 1.39 42,073,278 1986 4,932,725,468 1.71 85,142,8121960 3,039,669,330 1.33 40,792,172 1987 5,017,868,280 1.69 85,667,3321961 3,080,461,502 1.8 56,094,590 1988 5,103,535,612 1.66 85,671,9961962 3,136,556,092 2.19 69,516,194 1989 5,189,207,608 1.66 86,677,6811963 3,206,072,286 2.19 71,119,813 1990 5,275,885,289 1.58 83,940,3511964 3,277,192,099 2.08 69,031,982 1991 5,359,825,640 1.55 83,939,7111965 3,346,224,081 2.08 70,238,858 1992 5,443,765,351 1.48 81,404,0541966 3,416,462,939 2.02 69,755,364 1993 5,525,169,405 1.44 80,191,4341967 3,486,218,303 2.04 71,882,406 1994 5,605,360,839 1.43 80,626,2571968 3,558,100,709 2.08 74,679,905 1995 5,685,987,096 1.38 79,173,6611969 3,632,780,614 2.05 75,286,491 1996 5,765,160,757 1.37 79,745,1311970 3,708,067,105 2.07 77,587,001 1997 5,844,905,888 1.34 78,784,1751971 3,785,654,106 2.01 76,694,660 1998 5,923,690,063 1.31 78,308,5461972 3,862,348,766 1.95 76,183,283 1999 6,001,998,609 1.27 77,008,3731973 3,938,532,049 1.9 75,547,134 2000 6,079,006,982 1.23 75,318,8611974 4,014,079,183 1.81 73,265,577 2001 6,154,325,843 1.2 74,315,4601975 4,087,344,760 1.74 71,797,582 2002 6,228,641,303 1.18 73,845,3901976 4,159,142,342 1.72 72,213,985 2003 6,302,486,693 1.16 73,395,376

Table 1: World Population: 1950-2003, with growth rates

268

Appendix 2: Data Samples 269

Percent Avg. ann. Percent Avg. ann.Year Population grth. rate pop. change Year Population grth. rate pop. change2004 6,375,882,069 1.14 72,898,133 2027 7,956,320,645 0.74 58,800,5092005 6,448,780,202 1.12 72,714,711 2028 8,015,121,154 0.72 57,842,4662006 6,521,494,913 1.11 72,772,754 2029 8,072,963,620 0.7 56,862,5162007 6,594,267,667 1.1 72,776,911 2030 8,129,826,136 0.69 55,988,3832008 6,667,044,578 1.08 72,703,236 2031 8,185,814,519 0.67 55,209,4982009 6,739,747,814 1.07 72,485,099 2032 8,241,024,017 0.66 54,325,2042010 6,812,232,913 1.06 72,447,829 2033 8,295,349,221 0.64 53,388,4842011 6,884,680,742 1.05 72,509,964 2034 8,348,737,705 0.63 52,432,5682012 6,957,190,706 1.03 72,261,423 2035 8,401,170,273 0.61 51,582,5992013 7,029,452,129 1.02 71,803,520 2036 8,452,752,872 0.6 50,825,1992014 7,101,255,649 1 71,144,072 2037 8,503,578,071 0.59 49,945,3282015 7,172,399,721 0.98 70,443,548 2038 8,553,523,399 0.57 49,018,5662016 7,242,843,269 0.96 69,755,219 2039 8,602,541,965 0.56 48,111,0362017 7,312,598,488 0.94 68,928,253 2040 8,650,653,001 0.55 47,314,0212018 7,381,526,741 0.92 67,997,557 2041 8,697,967,022 0.53 46,592,5832019 7,449,524,298 0.89 66,966,195 2042 8,744,559,605 0.52 45,712,3042020 7,516,490,493 0.87 65,973,432 2043 8,790,271,909 0.51 44,766,0652021 7,582,463,925 0.85 65,024,404 2044 8,835,037,974 0.5 43,861,1932022 7,647,488,329 0.83 63,958,545 2045 8,878,899,167 0.48 43,037,0872023 7,711,446,874 0.81 62,831,316 2046 8,921,936,254 0.47 42,237,8242024 7,774,278,190 0.79 61,670,133 2047 8,964,174,078 0.46 41,228,8722025 7,835,948,323 0.77 60,634,107 2048 9,005,402,950 0.44 40,089,8862026 7,896,582,430 0.75 59,738,215 2049 9,045,492,836 0.43 39,002,569

2050 9,084,495,405

Table 2: World Population (projected): 2004-2050


The following tables gives data on malnutrition from UNICEF.138 UNICEF uses three differentmeasures for malnutrition, namely percentage of children underweight (UWT), percentage of chil-dren suffering from stunting, and percentage of children suffering from wasting. For each of thesethree variables, UNICEF gives two percentages. The first, denoted combined in the tables, is thepercentage who are classified as moderate or severe cases. The second percentage gives only thesever category.

UWT UWT Stunting Stunting Wasting WastingCountry combined severe combined severe combined severeAfghanistan 48 52 25Albania 14.3 4.3 31.7 17.3 11.1 3.6Algeria 6 1.2 18 5.1 2.8 0.6Armenia 2.5 0.1 13.6 2.7 2 0.3Azerbaijan 16.8 4.3 19.6 7.2 7.9 1.9Bahrain 8.7 1.8 9.7 2.7 5.3 0Bangladesh 47.8 13.1 44.8 18.4 10 1Benin 29.2 7.4 25 7.8 14.3 2.7Bhutan 18.7 3.1 40 14.5 2.6 0.5Bolivia 9.5 1.7 25.6 8.9 1.8 0.5Bosnia & Herz. 4.1 0.6 9.7 2.9 6.3 1.9Botswana 12.5 2.4 23.1 7.9 5 1.1Brazil 5.7 0.6 10.5 2.5 2.3 0.4Burkina Faso 34.3 11.8 36.8 16.6 13.2 2.5Burundi 45.1 13.3 56.8 27.7 7.5 0.5Cambodia 45.9 13.4 46 22 15.3 4Cameroon 21 4.2 34.6 13.3 4.5 0.8Cape Verde 13.5 1.8 16.2 4.8 5.6 1Central Afr. Rep. 24.3 6 38.9 19.1 8.9 2.1Chad 27.6 9.8 28.3 13.4 11.7 2.9Chile 0.8 1.9 0.3China 9.6 16.7 2.6Colombia 6.7 0.8 13.5 2.8 0.8 0.1Comoros 25.4 8.5 42.3 23.3 11.5 3.7Congo 13.9 3 18.8 6.6 3.9 0.9Congo, Dem. Rep. 34.4 10.2 45.2 24.6 9.6 3.5Costa Rica 5.1 0.4 6.1 2.3Cote d’Ivoire 21.4 4 21.9 7.8 10.3 0.9Croatia 0.6 0.8 0.8Cuba 4.1 0.4 4.6 1.1 2 0.4Czech Rep. 1 0 1.9 0.4 2.1 0.2Djibouti 18.2 5.9 25.7 13.2 12.9 2.8

Table 3: UNICEF Malnutrition Rates

138Data available online at www.childinfo.org/eddb/malnutrition/index.htm


UWT UWT Stunting Stunting Wasting WastingCountry combined severe combined severe combined severeDominican Rep. 4.6 1 6.1 1.6 1.5 0.1Ecuador 14.8 1.9 27.1 7.8Egypt 11.7 2.8 24.9 10.2 6.1 1.8El Salvador 11.8 0.8 23.3 5.7 1.1 0.1Eritrea 43.7 17 38.4 18.3 16.4 3.1Ethiopia 47.1 16 51.2 25.9 10.7 1.4Fiji 7.9 0.8 2.7 1.1 8.2 0.5Gambia 17 3.5 18.7 5.9 8.6 1.2Georgia 3.1 0.2 11.7 3.7 2.3 0.5Ghana 24.9 5.2 25.9 9.3 9.5 1.4Guatemala 24.2 4.7 46.4 21.2 2.5 0.9Guinea 23.2 5.1 26.1 10.1 9.1 2.1Guinea-Bissau 23.1 5.2 28 10.2 10.1 2.3Guyana 11.8 10.1 11.5Haiti 27.5 8.1 31.9 14.9 7.8 1.5Honduras 24.5 4 38.5 13.6 1.5 0.1Hungary 2.2 0.2 2.9 0.4 1.6 0.1India 47 18 45.5 23 15.5 2.8Indonesia 26.4 8.1Iran 10.9 1.5 15.4 3.8 4.9 0.9Iraq 15.9 22.1Jamaica 3.9 3.4 3.6Jordan 5.1 0.5 7.8 1.6 1.9 0.2Kazakstan 4.2 0.4 9.7 2.5 1.8 0.2Kenya 22.7 6.5 37.2 17.6 6.3 1.4Korea, Dem. 60 59.5 18.7Kuwait 9.8 2.9 23.8 11.8 10.6 2.7Kyrgyzstan 11 1.7 24.8 6 3.4 0.7Lao PDR 40 12.9 40.7 20.3 15.4 3Lebanon 3 0.3 12.2 2.9 2.9Lesotho 16 4 44 20 5 3Libya 4.7 0.6 15.1 4.5 2.8 0.4Madagascar 33.1 11.1 48.6 26 13.7 4.6Malawi 25.4 5.9 49 24.4 5.5 1.2Malaysia 18.3 1.2Maldives 43.2 10.1 26.9 6.8 16.8 3.4Mali 43.3Mauritania 23 9.2 44 32.8 7.2 3.2Mauritius 16.4 2.2 9.6 2.5 15 3.5Mexico 7.5 1.2 17.7 5.6 2 0.6Moldova 3.2 9.6 2.5Mongolia 12.7 2.8 24.6 8.5 5.5 1.2



UWT UWT Stunting Stunting Wasting WastingCountry combined severe combined severe combined severeMorocco 9 1.8 22.6 8 2.3 0.4Mozambique 26.1 9.1 35.9 15.7 7.9 2.1Myanmar 36 8.7 37.2 14.6 9.7 1.5Namibia 26.2 5.7 28.4 8.3 8.6 1.5Nepal 47.1 12 54.1 22.1 6.7 0.5Nicaragua 12.2 1.9 24.9 9.2 2.2 0.5Niger 39.6 14.3 39.8 19.5 14.1 3.2Nigeria 27.3 10.7 45.5 25.6 12.4 4.9Occupied Pal. Territ. 4.4 7.2 2.7Oman 23.6 3.9 22.9 8 13 1.6Pakistan 38.2 12.8Panama 6.8 14.4 1.1Paraguay 5 10.9 1Peru 7.8 1.1 25.8 8 1.1 0.3Philippines 28.2 29.9 5.6Qatar 5.5 8.1 1.5Romania 5.7 0.6 7.8 1.8 2.5 0.3Russian Fed. 3 0.5 12.7 6.7 3.9 1.6Rwanda 29 7.1 42.7 18.6 6.7 1.3Sao Tome & Principe 16 5 26 13 4.8Saudi Arabia 14.3 2.8 19.9 6.8 10.7 2.2Senegal 18.4 4.1 19 6 8.3Sierra Leone 27.2 8.7 33.9 15.8 9.8 1.9Somalia 25.8 6.9 23.3 12.1 17.2 3.5Sri Lanka 33 4.8 17 3.5 15Sudan 16.7 7.2Syria 12.9 3.7 20.8 10.1 8.7 2.5Tanzania 29.4 6.5 43.8 17.1 5.4 0.6TFYR Macedonia 6 0.7 6.9 1.7 3.6 0.5Thailand 18.6 16 5.9Togo 25.1 6.7 21.7 7 12.3 2.1Tunisia 4 0.6 12.3 3.4 2.2 0.5Turkey 8.3 1.4 16 6.1 1.9 0.4Uganda 25.5 6.7 38.3 15 5.3 0.9Ukraine 3 0.5 15.4 6.1 6.4 1.3United Arab Emirates 14.4 3.2 16.7 6.8 15.2 3.8United States 1.4 0.1 2.1 0.4 0.6 0.1Uruguay 4.5 0.8 7.9 1Uzbekistan 18.8 5 31.3 14 11.6 2.8Venezuela 4.7 0.7 13.6 4.5 3.1 0.6Vietnam 33.1 5.8 36.4 11.9 5.6Yemen 46.1 14.5 51.7 26.7 12.9 2.6Yugoslavia 1.9 0.4 5.1 1.9 3.7 0.7Zambia 25 59 4Zimbabwe 13 1.5 26.5 9.4 6.4 1.6



Democracy Scores and Literacy RatesThe Democracy Scores are from the Polity IV Democracy Project. Literacy rates are the

percentage of the population in the country that are considered functionally literate.

Country Dem. Lit. Country Dem. Lit.BELARUS 0 98.7 TUNISIA 1 27.4

UZBEKISTAN 0 96.7 TOGO 1 22.1KAZAKHSTAN 0 95.8 HAITI 1 22.0

CUBA 0 89.3 ALGERIA 1 21.5VIET NAM 0 83.0 BURUNDI 1 20.2MYANMAR 0 69.8 NEPAL 1 16.4

QATAR 0 58.2 YEMEN 1 14.2ZIMBABWE 0 57.6 CHAD 1 9.3

KUWAIT 0 57.6 TAJIKISTAN 2 90.7CHINA 0 52.9 SINGAPORE 2 72.9U.A.R. 0 52.2 JORDAN 2 55.1

BAHRAIN 0 50.9 BURKINA FASO 2 7.0SWAZILAND 0 48.8 CAMBODIA 3 48.9

EQUATORIAL GUINEA 0 46.0 ZAMBIA 3 47.7SYRIAN ARAB REP. 0 41.1 UNITED REP. OF TANZANIA 3 35.6

LAO P.D.R. 0 39.4 DJIBOUTI 3 30.2UGANDA 0 36.4 LIBERIA 3 18.4

LIBYA 0 35.4 ETHIOPIA 3 12.9SAUDI ARABIA 0 33.3 MALAYSIA 4 58.1

CONGO 0 32.9 COMOROS 4 49.5EGYPT 0 31.6 IRAN ISLAMIC REP. OF 4 34.3

ERITREA 0 29.0 NIGERIA 4 20.1RWANDA 0 27.8 NIGER 4 5.7

IRAQ 0 27.1 COTE D’IVOIRE 5 21.0MAURITANIA 0 26.8 CENTRAL AFRICAN REP. 5 14.0

SUDAN 0 24.8 GUINEA-BISSAU 5 12.0PAKISTAN 0 20.9 ARMENIA 6 91.6MOROCCO 0 19.8 GUYANA 6 90.7

OMAN 0 18.5 VENEZUELA 6 76.3GAMBIA 0 9.7 ECUADOR 6 74.3

CAMEROON 1 29.8 FIJI 6 73.2

Table 6:


COUNTRY Dem. Lit. COUNTRY Dem. Lit.NAMIBIA 6 57.0 GUATEMALA 8 45.1MALAWI 6 37.9 KENYA 8 40.6

BANGLADESH 6 24.6 SENEGAL 8 14.7MOZAMBIQUE 6 16.6 POLAND 9 98.2

BENIN 6 10.9 BULGARIA 9 92.4MALI 6 9.1 CHILE 9 87.6

ESTONIA 7 99.8 THAILAND 9 80.2UKRAINE 7 98.7 PANAMA 9 79.3

RUSSIAN FED. 7 98.2 PERU 9 71.5CROATIA 7 91.3 SOUTH AFRICA 9 69.7

SRI LANKA 7 80.5 JAMAICA 9 68.1PARAGUAY 7 79.8 BOLIVIA 9 57.5COLOMBIA 7 77.8 BOTSWANA 9 46.1

EL SALVADOR 7 57.9 INDIA 9 33.1ALBANIA 7 51.5 SLOVENIA 10 99.3

HONDURAS 7 50.6 LITHUANIA 10 98.3MADAGASCAR 7 38.5 HUNGARY 10 98.1

GHANA 7 29.5 MONGOLIA 10 95.2LATVIA 8 99.8 ITALY 10 94.5

ROMANIA 8 93.4 URUGUAY 10 93.3ARGENTINA 8 93.0 NETH. ANTILLES 10 93.0MOLDOVA 8 91.0 SPAIN 10 91.5

KOREA, REPUBLIC OF 8 86.8 TRINIDAD and TOBAGO 10 91.0PHILIPPINES 8 81.8 COSTA RICA 10 88.2

MEXICO 8 73.5 GREECE 10 86.5BRAZIL 8 68.4 CYPRUS 10 83.4

DOMINICAN REP. 8 67.2 ISRAEL 10 79.8LESOTHO 8 64.1 PORTUGAL 10 73.7TURKEY 8 56.5 MAURITIUS 10 67.0

INDONESIA 8 56.1 PAPUA NEW GUINEA 10 39.8NICARAGUA 8 54.5 Czech Rep 10 22.8

Table 7:


Social Capital Indices

State SCI State SCI State SCIAlabama -1.07 Maine 0.53 Ohio -0.18Arizona 0.06 Maryland -0.26 Oklahoma -0.16Arkansas -0.50 Massachusetts 0.22 Oregon 0.57California -0.18 Michigan 0.00 Pennsylvania -0.19Colorado 0.41 Minnesota 1.32 Rhode Island -0.06

Connecticut 0.27 Mississippi -1.17 South Carolina -0.88Delaware -0.01 Missouri 0.10 South Dakota 1.69Florida -0.47 Montana 1.29 Tennessee -0.96Georgia -1.15 Nebraska 1.15 Texas -0.55Idaho 0.07 Nevada -1.43 Utah 0.50Illinois -0.22 New Hamp. 0.77 Vermont 1.42Indiana -0.08 New Jersey -0.40 Virginia -0.32Iowa 0.98 New Mexico -0.35 Washington 0.65

Kansas 0.38 New York -0.36 West Virginia -0.83Kentucky -0.79 North Carolina -0.82 Wisconsin 0.59Louisiana -0.99 North Dakota 1.71 Wyoming 0.67

Table 1: Values for Putnam’s Social Capital Index

Sources for Data Online

• The Population Reference Bureau (http://www.prb.org/). Contains reports, articles, anddata related to population and demographics. Most of the data relates to recent years.

• United Nations Population Division (http://esa.un.org/unpp/). Time series data availableon population and several other variables by country.

• Poverty and inequality data from the Inter-American Development Bank (http://www.iadb.org/sds/pov/)

• UNICEF site, “Monitoring the Situation of Children and Women” (http://www.childinfo.org/).Includes malnutrition data.

• The Food and Agriculture Organization of the United Nations (www.fao.org). Site includesstatistical database with a large number of agricultural and nutritional variables.

• U.S. Census Bureau’s poverty site (http://www.census.gov/hhes/www/poverty.html).

Appendix 3: How Large is Large?

A billion here, a billion there, and pretty soon you’re talking real money—attributed to former Illinois Senator Everett McKinley Dirksen (1896-1969)

Big numbers pervade our society. There are billions of people in the world, millions in mostcountries. Government budgets can be in the trillions. In some sense, all of these numbers arebig numbers, and so it can seem like there is not much difference between them. Hopefully thefollowing will help you get some intuitive sense of how big a really big number really is.

1. Millions

(a) TIME: A million seconds is about 278 hours or about 11 and a half days. A millionminutes is about 694 days or almost 2 years. A million hours is about 114 years.

(b) A million dollars is enough to buy more than 30 brand new pick-up trucks. If you make$50,000 per year, it will take you 20 years to earn a total of a million dollars. A milliondollars is enough to run DWU for about a month.

2. Billions

(a) TIME: A billions seconds is 278,000 hours, or about 11,500 days or about 32 years. Abillion minutes is about 1900 years.

(b) MONEY:i. If you won the million dollar jackpot on Who Wants to be a Millionaire each and

every day for 2 years and 9 months, you would have won a total of a billion dollars.ii. If DWU had a billion dollars invested at 5% interest, it could use the interest to run

3 DWU’s forever, paying all the costs for everything and never ever charge a dimeof tuition.

iii. On average, the U.S. Defense Department spends a billion dollars every 36 hours.The U.S. Government spends a billion about every 4 hours.

3. Trillions

(a) TIME: A trillion seconds is about 32,000 years. A trillion minutes is about 1,900,000years. A trillion hours is about 120 million years, and that long ago, dinosaurs wouldhave been wandering the earth.

(b) MONEY:i. The annual U.S. budget is now over 2 trillion dollars.ii. If you wanted to spend a trillion dollars during your adult life, say from age 20

to age 80, you would have to spend about 45 million dollars a day, every day for60 years. If you did this by buying residential property, you could buy up all theresidential property in Mitchell in a few days. Buying all the residential propertyin Sioux Falls should take about 2 months or less. In a year, you could buy all theresidential property in South Dakota and be well into one of the neighboring states.

iii. On the other hand, if you wanted to spend a trillion dollars during your adult life,you could feed a lot of people. In many areas of the world, an adequate daily supplyof food can be purchased for a few cents. Even at 50 cents per person per day, youcould feed 90 million people, every day for 60 years. This is more than 10% of thetotal population of Africa.

(c) ECONOMICS:i. Annual U.S. Gross National Product (GNP) is about 10 trillion. This is about 20%

of the total world annual GNP.

276

Appendix 3: How Large is Large? 277

(d) SCIENCE:i. The total number of fish in the whole world is estimated to be about 1 trillion.ii. The number of cells in a person is about 50 trillion.iii. One light year is about 6 trillion miles. The nearest star (other than our sun) is

about 26 trillion miles or a little over 4 light years away.(e) FOOD, AGRICULTURE, and HUNGER:

i. Current annual worldwide production of grains is about 4 trillion pounds. This isenough for about 650 pounds per person per year or a little under two pounds perday. This is more than enough to live on.

4. Quadrillions (one quadrillion is 1015 or 1,000,000,000,000,000)

(a) TIME: A quadrillion seconds is about 32 million years. A trillion minutes is about 1.9billion years. A trillion hours is about 120 billion years. The universe is estimated tobe only 15 billion years old.

(b) MONEY:i. The estimated market price of all the assets in the world is 1 quadrillion.ii. If you wanted to spend a quadrillion dollars during your adult life, say from age 20

to age 80, you would have to spend about 45 billion dollars a day, every day for60 years. You could buy all the residential property in South Dakota and be wellinto one of the neighboring states in about 7 or 8 hours. Alternatively, given theprevious item on world assets, you could buy all the assets in the world in the firstyear. Not much left to do after that.

5. Really large numbers

(a) The number of atoms in the sun is about 1057.(b) The estimated number of atoms in the universe is about 1078.(c) The volume of the universe in cubic inches is about 1084.

For a bigger list of big numbers, visit http://pages.prodigy.net/jhonig/bignum/indx.html. Afew of our examples here are from this site.

Algebra: Modeling Our World...6 In addition, this course is designed to provide more relevant...

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